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+page_content='Recursive characterisations of random matrix  ensembles and associated combinatorial objects  Anas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rahman  ORCID iD: 0000-0003-2317-6685  Doctor of Philosophy – Science  January, 2022  The School of Mathematics and Statistics  The University of Melbourne  Submitted in total fulfilment for the degree of  Doctor of Philosophy – Science  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11428v1  [math-ph]  26 Jan 2023  Abstract  Mixed moments and cumulants of random matrices have been studied extensively over the  last half-century with applications in a large variety of fields ranging from enumerative geometry  to quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is particularly interesting to expand the cumulants in 1/N, where N  denotes matrix size, and study the coefficients of these expansions along with their generating  functions, which we call correlator expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We give an overview of the recursive  characterisations of random matrix ensembles that are currently at the forefront of random matrix  theory by way of studying two classes of ensembles using two different types of recursive schemes:  Established theory on Selberg correlation integrals is used to derive linear differential equations  on the eigenvalue densities and resolvents of the classical matrix ensembles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' which lead to 1-point  recursions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' understood to be analogues of the Harer–Zagier recursion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for the expansion coefficients  of the associated 1-point cumulants,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' while loop equation analysis is used to recursively compute some  leading order correlator expansion coefficients pertaining to certain products of random matrices  that have recently come into interest due to their connections to Muttalib–Borodin ensembles and  integrals of Harish-Chandra–Itzykson–Zuber type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We also show how the aforementioned differential  equations can be used to characterise the large N limiting statistics of the classical matrix ensembles’  eigenvalue densities in the global and edge scaling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A common theme between the two classes of ensembles studied in this thesis is that their  representative random matrices can, for the most part, be constructed from Ginibre matrices (matrices  whose entries are independently and identically distributed normal variables), which allows for  their mixed moments and cumulants to be interpreted as enumerations of particular ribbon graphs,  topological and combinatorial maps, and/or topological and combinatorial hypermaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A major  part of this thesis is devoted to a comprehensive review of how the Isserlis–Wick theorem implies  these interpretations for the mixed moments and cumulants of the Gaussian and Laguerre ensembles,  which leads on to original discussion on how the relevant theory extends to the matrix products  mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the loop equations derived in this thesis have the added value of solving  certain problems in enumerative combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In order to make this thesis self-contained and to properly motivate our original contributions,  a decent portion of our development constitutes a pedagogically detailed survey of the contiguous  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is therefore expected that this thesis will serve as a valuable resource for readers  wanting a well-rounded introduction to classical matrix ensembles, matrix product ensembles, 1-point  recursions, loop equations, Selberg correlation integrals, and ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  i  Declaration  This is to certify that:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the thesis comprises only my original work towards the degree of Doctor of Philosophy  except where indicated in the preface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' due acknowledgement has been made in the text to all other material used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the thesis is fewer than 100 000 words in length, exclusive of bibliographies, tables,  maps, and appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Anas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rahman  January 22, 2022  ii  Preface  The introductory content of Chapter 1 is wholly original to this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' So too are the contents  of Chapter 2 preceding §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, whereas the content of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 is drawn from the work “Linear  differential equations for the resolvents of the classical matrix ensembles” [279] done by  the present author in collaboration with Peter Forrester and published by Random Matrices:  Theory and Applications in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 contains a mild enhancement of content from the  work “Relations between moments for the Jacobi and Cauchy random matrix ensembles”  [130] done by the present author in collaboration with Peter Forrester and published by the  Journal of Mathematical Physics in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The contribution of the present author towards the  works [279], [130] is equal to that of Forrester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thereafter, the contents of Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 are amalgamations of content from  [279], [130] reworked slightly for coherence and supplemented with additional, original  results for completeness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Appendix B is taken from [279].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3,  Appendix C, and Appendix D, we give a heavily detailed presentation of the ongoing work  “Combinatorics and loop equations for antisymmetrised and Hermitised matrix product  ensembles” [75] of the present author and Stephane Dartois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The results pertaining to [75] are  born of vigorous discussions between the present author and Dartois, but the corresponding  outcomes displayed in this thesis are due solely to the present author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The 2017 article [142] by the present author, Peter Forrester, and Nicholas Witte is briefly  reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, but is work mostly completed prior to commencement of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus,  it is cited appropriately where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' All other contents of this thesis not addressed in  the above are original to this thesis and have not been published elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The candidature of the author was financially supported by the Australian Government  Research Training Program Scholarship, the ARC Centre of Excellence for Mathematical and  Statistical Frontiers, and the ARC Grant DP210102887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  iii  Acknowledgements  First and foremost, I would like to thank my supervisors Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Forrester, Mario Kieburg,  and Paul T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Norbury for the many years of patient guidance through the various topics  encountered in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, Peter’s uncanny ability to recall obscure tidbits from  his vast knowledgebase and provide insightful feedback on written text combined with his  generous willingness to do so upon request at almost any time of day, regardless of frequency,  has most certainly been a privilege to experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It would have been impossible for me  to learn the amount of random matrix theory that I did in the last few years without the  countless hours of effort put in by Peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, I would like to thank my co-supervisors  Mario and Paul for introducing me to topics in analysis and geometry, respectively, which  have contributed to my growth as a mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Next, I would like to thank those who would explain things to me for hours on end or  just lend me a friendly ear without any obligation to do so: My various interactions with  Stephane Dartois, Jesper Ipsen, Shi-Hao Li, Anthony Mays, Norman Do, David Ridout, Nora  Ganter, Thomas Quella, Arun Ram, Paul Fijn, and Gufang Zhao have enlightened me on  the value and sheer joy of belonging to an academic community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, Stephane’s  skill in explaining intricate topics in a sympathetic and illuminating manner has made it an  absolute pleasure to collaborate with him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As for peers with whom I shared the experience of being a graduate student, I am  grateful to Allan Trinh, Jiyuan Zhang, Srivatsa Badariprasad, Wee Chaimanowong, Eric  Shen, Campbell Wheeler, and Behrooz Niknami for years of friendship and interesting  conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is not unusual to spend a majority of your graduate studies in solitude,  especially when working in such a specialised field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Therefore, I consider myself very lucky  to have shared an office with two friends, Allan and Jiyuan, with whom I could discuss  random matrix theory to my heart’s content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, conversations with friends  from outside random matrix theory have given me opportunities to take a break from  thinking about my work while affording me a chance to learn about interesting topics far  removed from my field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally on the academic side, I would like to note that my mathematical maturity has  benefitted greatly from the seminars, conferences, and summer schools that I have been able  iv  to participate in — I am indebted to the people who make these events possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In that vein,  I am similarly indebted to the academic support staff at our ACEMS node and the School  of Mathematics and Statistics for their timely help with all administrative issues, such as  sorting out IT access, arranging travel funds, or organising seminars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In terms of non-academic support, I would like to thank my friends and family for their  love and support, without which I could not have completed this journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' I am grateful to  my mother for instilling in me the confidence to pursue my passions from a young age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To  my family and the friends that I have not been able to see very often, I apologise and thank  you for your understanding of my absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is in the nature of time for it to grow scarce  and it is truly a shame whenever one must spend it alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Most of all, my utmost gratitude is reserved for my beautiful fianc´ee Johanna, who has  quelled my worries in my times of woe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' who has shown seemingly blind faith in my abilities  when I had little faith of my own;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' who has been understanding at times when I have been  unavailable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and who has been the source of my first smile of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thank you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  v  Contents  Abstract  i  Declaration  ii  Preface  iii  Acknowledgements  iv  1  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Random Matrix Ensembles  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  Quantities of interest .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='2  Classical Matrix Ensembles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  The Cauchy and circular Jacobi ensembles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  Selberg Correlation Integrals  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  The Selberg and Dixon–Anderson integrals  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='  98  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  Global scaled differential equations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='2  Soft and hard edge scaled differential equations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='  109  3  Characterisations of the Moments and Cumulants  117  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Recurrence Relations for the Moments of the Classical Matrix Ensembles  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  Recurrences for the spectral moments .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='2  1-point recursions for the moment expansion coefficients  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='2  Established Results on the Moments of the Classical Matrix Ensembles  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1  Results from skew-orthogonal polynomial theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='2  Results from symmetric function theory  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='3  Results in terms of hypergeometric orthogonal polynomials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  243  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  The topological hypermaps corresponding to limN→∞ ˜c(J1)  2,2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  244  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5  The topological hypermaps corresponding to c(H1),0  4  , c(H1),2  2  , and c(H1),0  1,1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  256  xi  Chapter 1  Introduction  The overarching theme of this thesis is that of recursive structures within random matrix  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The focus is on two types of recursions: 1-point recursions for moment expansion  coefficients and loop equations for n-point correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A method of deriving 1-point recursions  from differential equations satisfied by the eigenvalue densities of the classical matrix ensembles  is demonstrated in Chapters 2 and 3, while the method of loop equations is demonstrated  through an application to certain matrix product ensembles in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These two classes  of random matrix ensembles are respectively introduced in Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, along with  auxiliary details drawn from the literature so as to give our development sufficient context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Before introducing these ensembles, we review some pertinent fundamentals of random  matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Random Matrix Ensembles  Borrowing language from statistical mechanics, an ensemble can be thought of as a virtual  collection of all possible states that a given system can be in, with more probable states  appearing more often in the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, if one draws uniformly from the ensemble,  one is more likely to retrieve more probable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One way to obtain such an ensemble is to  generate copies of the system of interest ad infinitum, with the final infinite collection being  named the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With this intuition in mind, we henceforth work with the following  rigorous definition, in the case that the underlying system is a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A random matrix ensemble E = (S, P) is a set S of matrices of fixed size  equipped with a probability density function (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') P : S → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To say that a matrix  X is drawn from the random matrix ensemble E is to say that it is a random variable with  values in S and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is commonplace to define a random matrix ensemble by prescribing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s  Pij on the independent entries Xij of a representative random matrix X ∈ E along with a  specification of how the other entries of X depend on these independent entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In such  cases, the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' on S is the joint probability density function (j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') formed by taking the  product  P(X) =  ∏  i,j s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' {Xij} ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Pij(Xij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 allows for a broad range of examples: For many random matrix ensembles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  the set S of say M × N matrices is described by applying constraints to the set of real  matrices MM×N(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the set of complex matrices MM×N(C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' or the set of quaternionic  matrices MM×N(H) (see Appendix A for an introduction to quaternionic matrices),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' though  the definition above does not forbid more exotic matrix entries like octonions [264],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [124] or p-  adic numbers [261],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' to name a few examples — the Frobenius theorem [146],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [97] directs our  attention to matrices with entries in R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' or H due to their being the only finite-dimensional  associative division algebras over the real numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The constraints  used to construct S are usually those that require the elements of S to exhibit features such  as orthogonality, unitarity, (skew-)symmetry, Hermiticity, positive definiteness, invariance  under complex conjugation, or combinations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, S is oftentimes simply the set of  symmetric, Hermitian, or skew-symmetric matrices, but in physical settings is more likely to  be some (possibly trivial) quotient of products of the classical matrix groups O(N), U(N),  and Sp(N), of which ten such amalgamations stand out from the viewpoint of modelling  Hamiltonians subject to symmetry constraints [24], [283];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' these are in correspondence with  the ten infinite families of matrix Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another type of constraint that has been  explored in addition to these is that of requiring entries Xij of the representative random  matrix X ∈ E to vanish when the N-periodic distance between i and j is greater than some  threshold called the band width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These so-called random band matrices relate to the study of  Anderson localisation-delocalisation transitions [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Random Matrix Ensembles  When it comes to determining the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P on S, there are again many options that  have garnered interest throughout the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There are far too many to cover here in  totality, but let us highlight the diversity of topics in random matrix theory through a few  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A simple yet non-trivial example is that of Bernoulli matrices B whose entries  Bij are all independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') Bernoulli random variables ±1:  S = MN×N(R) and Pij(Bij) = 1  2[δ(Bij − 1) + δ(Bij + 1)], where δ is the Dirac delta [208], [297].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  If we instead require that B be symmetric, its diagonal entries be zero, and its independent  hence upper triangular entries have p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Pij(Bij) = c  N δ(Bij − 1) +  �  1 − c  N  �  δ(Bij),  0 < c ≡ c(N) ≪ N,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  B belongs to the ensemble of Erd˝os–R´enyi matrices [103], [102], which are the adjacency  matrices of Erd˝os–R´enyi random graphs [104], [157];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the parameter c is the mean connectivity  of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This ensemble is a type of sparse random matrix ensemble [282], [214], [323], which  is an umbrella term for a variety of random matrix ensembles whose shared characteristic  is that for each such ensemble, the independent entries Bij of their representative random  matrix B (now allowed to be complex and/or have aforementioned constraints relaxed) have  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Pij of the form given in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) but with δ(Bij − 1) replaced by any p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of  interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This brings us to the question of which p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s are ‘interesting’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moving on from Bernoulli and sparse random matrix ensembles, let us consider a more  general applications-based perspective where one would like to model real world systems  such as ecologies [236], heavy atom spectra [313], [98], [316], wireless communication  channels [301], random quantum states [68], log-gases [119], or quantum transport [33],  among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From this viewpoint, where one chooses to eschew knowledge of the  finer details of the real world system at hand in favour of treating them statistically, there  are essentially three ideologies that dominate the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The first is to assume as little as  possible about the system and work in full generality, focusing on universal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this  line of thinking, one studies Wigner matrix ensembles [313], [315], [294] or, more recently,  general Wigner-type matrix ensembles [6]: Let X be drawn from either the real symmetric or  complex Hermitian N × N Wigner matrix ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, its entries Xij are independently  distributed for 1 ⩽ i ⩽ j ⩽ N, with mean zero and all moments having an upper bound  independent of i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In addition, the diagonal entries have identical variance while the upper  3  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  triangular entries have variance one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If we relax the condition on the variances, X is instead  of general Wigner-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Many random matrix ensembles fall into the class of Wigner matrix  ensembles due to the freedom enjoyed by the variables Xij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, results proved for  Wigner matrix ensembles hold when the matrix entries are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' centred normal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The second of the three dominant ideologies is that of studying uniform distributions,  which corresponds to a principle often seen in statistical mechanics: In the absence of any  relevant information, one should assign equal probability to all states of an ensemble (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the  microcanonical ensemble, dice, a deck of cards, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our discussion up to this point has  focused on random matrix ensembles distributed according to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P defined with respect  to the flat Lebesgue measure induced by the canonical embedding of S into Euclidean space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', for the Bernoulli matrix ensemble, we have implicitly taken  dB =  N  ∏  i,j=1  dBij,  while for the complex Hermitian Wigner matrix ensemble, we have taken  dX =  N  ∏  i=1  dXii  ∏  1⩽j<k⩽N  dX(1)  jk dX(2)  jk ,  Xjk = X(1)  jk + iX(2)  jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  It is not possible to define a uniform probability density with respect to such measures on  non-compact domains S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, one must consider compact groups S endowed with their  respective Haar (probability) measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The prime example is the circular unitary ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 (Haar, compact case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Given a compact group S, let U ∈ S be a generic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  There exists a unique (up to normalisation) measure dµ(U) on S, called the Haar measure (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  [171, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 11], [255, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5]), such that  �  S dµ(U) < ∞  and for any fixed U0 ∈ S,  dµ(U0U) = dµ(UU0) = dµ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  If dµ(U) integrates to unity on S, it is referred to as the Haar probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The Haar measure is uniform in the sense that for any Borel subset S′ ⊆ S and fixed  U0 ∈ S, the Haar volume of S′ is invariant under left and right translations of S′ by U0:  �  U0S′ dµ(U) =  �  S′U0  dµ(U) =  �  S′ dµ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Random Matrix Ensembles  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The N × N circular unitary ensemble (CUE) is the unitary group U(N) with  its Haar measure [97], [98], [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2] (made explicit in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If U ∈ CUE(N),  then the symmetric unitary matrix UTU represents the N × N circular orthogonal ensemble  (COE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we have S ≃ U(N)/O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If U ∈ CUE(2N), then UDU represents the N × N circular  symplectic ensemble (CSE), where  UD := J2N UT J−1  2N  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  is the quaternionic adjoint of U (for X ∈ MN×N(H) with the quaternion entries written as  2 × 2 complex matrices, we write X† for XD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Appendix A), with  J2N := IN ⊗  �  �0  −1  1  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  That is, J2N is the 2N × 2N block-diagonal matrix with non-zero blocks given by the 2 × 2  matrix on the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the CSE case, S ≃ U(2N)/Sp(2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The third and final ideology is that of working with algebraic combinations of matrices  whose entries are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' normal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' By specifying a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' on matrix entries and taking  such combinations, one is able to model finer details than would be possible with Wigner  matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The normal distribution is a natural choice for the entries due to the central  limit theorem and the fact that it is the simplest probability distribution on the real line, at  least in the sense of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More importantly, the normal distribution and, by extension,  this class of random matrix ensembles have been investigated extensively throughout the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, there are a plethora of tools available for performing concrete computations  and, consequently, the study of these types of random matrix ensembles is deeply connected  to topics throughout mathematics, sometimes in surprising ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is for these reasons  that the ensembles we choose to study in this thesis primarily belong to this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed,  connections to Selberg correlation integrals, ribbon graph combinatorics, and loop equations  act as the foundation of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us now formally introduce the building block from  which most of the ensembles of Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The M × N real, complex, and quaternionic Ginibre ensembles [158] are  respectively MM×N(R), MM×N(C), and MM×N(H) with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(Gin)(G) = π−MNβ/2 exp  �  − Tr G†G  �  =  M  ∏  i=1  N  ∏  j=1  π−β/2 exp  �−|Gij|2�  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  5  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  which is defined with respect to the Lebesgue measure  dG =  M  ∏  i=1  N  ∏  j=1  β  ∏  s=1  dG(s)  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  Here and henceforth, other than in the context of the circular ensembles, the Dyson index  β [97] counts the generic number of real components G(s)  ij  of each matrix entry Gij, so that  β = 1 corresponds to real ensembles, β = 2 corresponds to complex ensembles, and β = 4  corresponds to quaternionic ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As seen in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), it allows us to treat the  real, complex, and quaternionic cases simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In similar fashion to the circular ensembles defined earlier, it is customary for the N × N  real, complex, and quaternionic Ginibre ensembles to be referred to as the Ginibre orthogonal,  unitary, and symplectic ensembles (GinOE, GinUE, GinSE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The reasoning behind  this terminology is given in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Before using the Ginibre ensembles to construct the classical matrix ensembles and matrix  product ensembles that are considered in this thesis, we introduce the observables that will  be at the centre of our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Quantities of interest  The random matrices studied in this thesis (these are yet to be introduced and do not  include those defined above) are diagonalisable and self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, they are determined  by their eigenvalues and eigenvectors, the former of which are henceforth assumed to be  indistinguishable and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' While the eigenvectors of random matrices have been a topic of  study [269], eigenvalue statistics have attracted considerably more attention and will likewise  be the subject of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, in addition to the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P(X) of say N × N random  matrices X, we are also interested in the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) of the eigenvalues  {λi}N  i=1 of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is defined to be such that  �  [a1,b1]×···×[aN,bN] p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) dλ1 · · · dλN  is equal to the probability that ai ⩽ λi ⩽ bi simultaneously for all 1 ⩽ i ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the N × N circular ensembles is  p(C)(eiθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , eiθN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (C)  N,β  |∆N(eiθ)|β,  θi ∈ [0, 2π),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Random Matrix Ensembles  where N (C)  N,β = (2π)NΓ(βN/2 + 1)/[Γ(β/2 + 1)]N is a normalisation constant [98, I],  ∆N(λ) := Det  �  λj−1  i  �N  i,j=1 =  ∏  1⩽i<j⩽N  (λj − λi)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  is the Vandermonde determinant, and β = 1, 2, and 4 correspond to the COE, CUE, and  CSE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From here on out, we will not write out function arguments following  semicolons when their values are clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Related to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is the univariate eigenvalue density defined by  ρ(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) := N  �  RN−1 p(λ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) dλ2 · · · dλN,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  which is normalised such that the expected number of eigenvalues lying between a and b is  given by � b  a ρ(λ) dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that ρ(λ) is not a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for N ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To enable more concise notation, let us now introduce the ensemble average  ⟨O⟩ :=  �  RN {O p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN)} dλ1 · · · dλN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  It must be understood that O is to be interpreted as an operator here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, if  O =  ∂  ∂λ1 λ1, one should read the above equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) as “multiply the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) by λ1, differentiate the resulting product with respect to λ1, then integrate over  RN.” Using this notation, we can write the eigenvalue density (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) as the linear statistic  ρ(λ) =  N  ∑  i=1  ⟨δ(λ − λi)⟩  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  due to the assumed indistinguishability of the eigenvalues {λi}N  i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' if the eigenvalues were  distinguishable, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) would supersede (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) as the correct definition for the  eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' While the definition of the ensemble average (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) is given in terms of  the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN), we will also utilise this notation for ensemble averages  with respect to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The different uses will be distinguished through  the use of subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For the purposes of this thesis, the eigenvalue densities considered are fully characterised  by their spectral moments (one may, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', check Carleman’s condition [308, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 88])1  mk :=  �  R λkρ(λ) dλ,  k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  1A subtlety arises in the case of the Cauchy ensembles defined by equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) upcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Then, there is a need for analytic continuation to make sense of the moments which otherwise diverge for k  large enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  7  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Indeed, if we are interested in a linear statistic ∑N  i=1 f (λi) of the eigenvalues such that f (λ)  is analytic at zero with Taylor expansion f (λ) = ∑∞  k=0 fk λk, we can compute its expectation  to be  �  N  ∑  i=1  f (λi)  �  =  �  R f (λ)ρ(λ) dλ =  ∞  ∑  k=0  fk mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  The spectral moments can additionally be expressed as an average with respect to the  eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) and the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) of the random matrix X according to  mk =  �  N  ∑  i=1  λk  i  �  =  �  Tr Xk�  P(X) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  where we define  ⟨O⟩P(X) :=  �  S {O P(X)} dX  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  in analogy with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11), with S as given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  An invaluable tool for studying the spectral moments is the moment generating function  W1(x) :=  ∞  ∑  k=0  mk  xk+1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  which is also known as the resolvent due to it being equal to the Stieltjes transform [308,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 65] of the eigenvalue density:  W1(x) =  �  supp ρ  ρ(λ)  x − λdλ,  x ∈ C \\ supp ρ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  where supp ρ := {λ ∈ R | ρ(λ) ̸= 0} is the support of ρ (here, S denotes the closure of S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Obtaining a closed form expression for the resolvent allows one to recover the eigenvalue  density via the Sokhotski–Plemelj inversion formula [308, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 65]  ρ(λ) = lim  ε→0  W1(λ − iε) − W1(λ + iε)  2πi  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  so long as ρ(λ) is a priori known to be continuous on its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15),  we can express the resolvent as an ensemble average with respect to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) and the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) on S:  W1(x) =  �  N  ∑  i=1  1  x − λi  �  =  �  Tr  1  x − X  �  P(X)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  Being able to choose between three expressions for the resolvent is quite a boon, as lacking a  tractable form for the eigenvalue density is no longer a hindrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is exactly the case  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Random Matrix Ensembles  for the matrix product ensembles considered in Chapter 4, and the computations presented  there are in terms of the ensemble average ⟨ · ⟩P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, the resolvent is usually easier  to work with than the eigenvalue density, one reason for which being that the former often  admits a large N expansion in 1/N while the latter does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 (Borot–Guionnet ’12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Assuming certain hypotheses [41] on the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN), the resolvent W1(x) and, more generally, the soon to be defined connected n-point  correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) have large N expansions of the form  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) = N2−n  ∞  ∑  l=0  Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  Nl  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  where the correlator expansion coefficients Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) have no dependence on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, we use this theorem to study the large N behaviour of the classical matrix  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 does not apply directly to a subset of these ensembles  due to a technical issue that we must first resolve by scaling the ensembles so that their  eigenvalues lie within a compact connected set in the N → ∞ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In these cases, we  must study the global scaled eigenvalue density ˜ρ(λ) and scaled resolvent ˜W1(x) defined by  ˜ρ(λ) := cN  N ρ(cNλ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  ˜W1(x) := cNW1(cNx)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  = N  �  supp ˜ρ  ˜ρ(λ)  x − λdλ,  x ∈ C \\ supp ˜ρ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  where cN is a scaling parameter that differs from ensemble to ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (In the case of our  matrix product ensembles, we do not check the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and instead opt  to establish the existence of expansions of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) via combinatorial arguments  given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') Now, assuming that the scaled resolvent has an expansion of the form  ˜W1(x) = ∑∞  l=0 Wl  1(x)N1−l as given by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), use the Sokhotski–Plemelj inversion  formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) to define  ρl(λ) := lim  ε→0  Wl  1(λ − iε) − Wl  1(λ + iε)  2πi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  Then, ˜ρ(λ) and ρ0(λ) agree in the N → ∞ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, ˜ρ(λ) and ∑∞  l=0 ρl(λ)N−l do not  agree at finite N, as one might expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, the latter sum is referred to as the smoothed  9  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  eigenvalue density, so called due to it lacking oscillatory terms that are known to be present in  the true global scaled eigenvalue density ˜ρ(λ) [152], [140], [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless, the moments  of the smoothed eigenvalue density, computed via analytic continuation, agree with those  of ˜ρ(λ), due to the missing oscillatory terms being compensated for by non-integrable  singularities at the endpoints of support [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A consequence of this is that for l > 0,  the corrections ρl(λ) integrate to zero on their supports and are thus signed densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In  short, the following diagram does not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  ρ(λ)  ˜ρ(λ)  �  ρl(λ)  �∞  l=0  W1(x)  1  N ˜W1(x)  �  Wl  1(x)  �∞  l=0  scaling cN  Stieltjes transform  Stieltjes transform  moments  Stieltjes transform  scaling cN  Sokhotski–Plemelj  Sokhotski–Plemelj  1/N expansion  Sokhotski–Plemelj  Let us now turn our attention to Chapter 4, wherein we derive three types of loop  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The first set of loop equations are satisfied by the unconnected n-point correlators  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) :=  �  n  ∏  i=1  Tr  1  xi − X  �  P(X)  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  which act as generating functions for the mixed moments  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn :=  �  n  ∏  i=1  Tr Xki  �  P(X)  ,  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  Related to the mixed moments are the mixed cumulants cκi, which are defined implicitly by  the moment-cumulants relation [237, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2]  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  ∑  K⊢{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn} ∏  κi∈K  cκi,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  where K ⊢ {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn} means that K is a partition of {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', K = {κi}m  i=1 for some  1 ⩽ m ⩽ n such that the disjoint union κ1 ⊔ · · · ⊔ κm is equal to {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These mixed  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Random Matrix Ensembles  cumulants are generated by the so-called connected n-point correlators  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) :=  ∞  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn=0  ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  xk1+1  1  · · xkn+1  n  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  =  ∑  G⊢{x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',xn}  (−1)#G−1(#G − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ∏  Gi∈G  U#Gi(Gi)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  = Un(x1, Jn) − ∑  ∅̸=J⊆Jn  Wn−#J(x1, Jn \\ J)U#J(J),  Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  where #S denotes the size of the set S [289], [119, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 187], [320, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 8–9]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Setting n = 1, 2  in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) shows that  W1(x) = U1(x),  W2(x1, x2) = CovP(X)  �  Tr  1  x1 − X, Tr  1  x2 − X  �  ,  where  CovP(X)( f (X), g(X)) :=  �  ( f (X) − ⟨ f (X)⟩P(X))(g(X) − ⟨g(X)⟩P(X))  �  P(X)  denotes the covariance of the linear statistics f (X) and g(X) with respect to P(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a similar  formula is found for n = 3, but the analogous structures for n ⩾ 4 are more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) is used in Chapter 4 to transform the set of loop equations on the un-  connected correlators Un into a second set of loop equations satisfied by the connected  correlators Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Our interest in the connected correlators Wn over their unconnected counterparts Un is  that the former is of lower order in N due to the cancelling of leading order terms brought on  by the inclusion-exclusion structure of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, the connected correlators Wn  have the advantage of admitting large N expansions of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is significant  because the loop equations on both the connected and unconnected correlators fail to close  and are thus unsolvable, but combining them with the large N expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) yields a  third set of loop equations for the correlator expansion coefficients Wl  n that form a triangular  recursive system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Considering n = 1 for simplicity, this means that Wl  1 can be computed  through (at most)  �(1 + l/2)2�  applications of the loop equations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [142]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Although  complexity grows strongly with l, it is quite feasible to write down W0  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , W3  1 and thus  calculate the scaled spectral moments up to a corresponding order in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be precise,  2The otherwise formal series (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) converges when |x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , |xn| >  max  λ∈supp ρ|λ|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  characterisations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) in terms of the Un follow from (the inverse of) the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  11  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  let ˜mk denote the spectral moments of ˜ρ(λ) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the scaled spectral moments of ρ(λ)) and  implicitly define the moment expansion coefficients ˜Mk,l according to  ˜mk =  ∞  ∑  l=0  ˜Mk,l N−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  Then, if Coeff(x, k) denotes the action of expanding in 1/x and extracting the coefficient of  1/xk, the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  ˜W1(x)  Wl  1(x)  ˜mk  ˜Mk,l  Coeff(N, l)  Coeff(x, k + 1)  Coeff(x, k + 1)  Coeff(N, l)  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We recognise that the use of the adjectives ‘unconnected’ and ‘connected’ has yet  to be motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This terminology alludes to the fact that the mixed moments and cumulants  often relate to problems of counting topological surfaces that are respectively unconnected  or connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we address the details of this relationship in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Having defined the observables of interest in full generality, it is now time to introduce  the specific forms that are relevant to the contents of Chapters 2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Classical Matrix Ensembles  Let us first define the classical matrix ensembles that will be at the focus of Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, let G be drawn from the N × N real, complex, or  quaternionic Ginibre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the random matrix H = 1  2(G† + G) represents the  corresponding Gaussian (also known as Hermite) ensemble [314], [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If we instead take G to  be M × N Ginibre, then W = G†G is said to be drawn from the (M, N) Laguerre (also known  as Wishart) ensemble [317].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We say that Y is an element of the (M1, M2, N) real, complex, or  quaternionic Jacobi ensemble if it is of the form Y = W1(W1 + W2)−1 where the Wi belong to  the corresponding (Mi, N) Laguerre ensembles with Mi ⩾ N [249, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  The real Gaussian ensemble was first introduced as a statistical model for heavy atom  spectra by Wigner [314] in 1957, while its complex and quaternionic counterparts were  introduced by Dyson [98] in 1962 with the same application in mind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in this latter work,  Dyson also highlighted, and made application of, analogies with the statistical mechanics  of log-gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These and related papers from the surrounding literature are conveniently  collated and reviewed in the 1965 book [275] of Porter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Laguerre (Wishart) ensemble,  on the other hand, has been a mainstay of multivariate statistical analysis since the 1928  work [317] of Wishart showing how to change variables from the entries of an M × N real  Ginibre matrix G to the entries of the Wishart–Laguerre matrix W = GTG defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The textbook [249] details applications of the real and complex Laguerre ensembles in  multivariate statistics, including its role in estimating covariance matrices and supplying null  hypotheses for principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Around the turn of the century, the Laguerre  ensembles found further applications in the fields of wireless communications [301] and  quantum transport [33] (one such application is briefly discussed early in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  review [33] also outlines how problems of quantum transport served as early motivation  for studying the Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another appreciable motivation was given a decade  later in the 2008 work [197] of Johnstone showing how the eigenvalue statistics of the Jacobi  ensembles can be used to determine null hypotheses for multivariate analysis of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The matrices given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 are all self-adjoint and thus have N real eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Taking either the real, complex, or quaternionic case for definiteness, the entries of the  Hermite–Gaussian random matrix H are independent (up to the constraint of H being  self-adjoint) centred normal variables with the diagonal entries having variance 1  2 and the  real components of the upper triangular entries having variance 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, it can be observed  that the Gaussian ensembles are examples of Wigner matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The diagonal entries  of the Wishart–Laguerre random matrix W are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' chi-squared variables with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(L)  ii (Wii) =  1  Γ(βM/2)WβM/2−1  ii  e−Wii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the off-diagonal entries of W and indeed all entries of the Jacobi random matrix  Y are too complicated to present here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the contrary, the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the random matrices  H, W, and Y are themselves quite elegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  13  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let κ := β/2, a = κ(M1 − N + 1) − 1, and b = κ(M2 − N + 1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From the  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) of the Ginibre ensembles, we can change variables to obtain the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the N × N  Gaussian, (M1, N) Laguerre, and (M1, M2, N) Jacobi ensembles, respectively [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1,3]:  P(G)(H) =  1  Z(G)  N,β  exp  �− Tr H2�  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  P(L)(W) =  1  Z(L)  N,a  (Det W)a exp (− Tr W) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  P(J)(Y) =  1  Z(J)  N,a,b  (Det Y)a (Det(IN − Y))b ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  where the ZN are normalisation constants, often referred to as partition functions, whose explicit  forms are inconsequential to us but can be obtained from the method described at the end of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With β = 1, 2, 4 corresponding to F = R, C, H, the random matrix ensembles  introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 are specified by assigning the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) to the matrix  sets  S(G) = {H ∈ MN×N(F) | H = H†},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  S(L) = {W ∈ S(G) | W positive definite},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  S(J) = {Y ∈ S(L) | IN − Y positive definite},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Apart from the matrix Y given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, there are some other important  constructions that are known to yield random matrices Y′, Y′′, Y′′′, which all have p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  with a, b as prescribed in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 [119, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6], [330], [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let X be a Haar-distributed element of either O(M1 + M2) (β = 1), U(M1 + M2)  (β = 2), or Sp(2(M1 + M2)) (β = 4) for positive integers M1, M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Take T to be the  M1 × N truncation of X with Tij = Xij for 1 ⩽ i ⩽ M1, 1 ⩽ j ⩽ N, and M1, M2 ⩾ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Define Y′ = T†T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For i = 1, 2, let Gi be Mi × N Ginibre matrices with Mi ⩾ N and define  X =  �  �G1  G2  �  � ,  ˜I =  �  �IM1  0M2  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Set Y′′ = X† ˜IX(X†X)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the notation of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, let Y′′′ := (W1 + W2)−1/2W1(W1 + W2)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The term ‘classical matrix ensemble’ originally referred to the Gaussian, Laguerre and  Jacobi ensembles due to their relation to the classical orthogonal polynomials, which we  briefly review in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, the family of classical matrix ensembles has one more  constituent according to the contemporary definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A random matrix ensemble is said to be a classical matrix ensemble [119, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3],  [138] if its eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is of the form  p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (w)  N,β  N  ∏  i=1  w(λi) |∆N(λ)|β,  β = 1, 2, 4,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  where N (w)  N,β is a normalisation constant with values given in [93], [129] (alternatively, see  §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), ∆N(λ) is the Vandermonde determinant, as specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), and the  weight function w(λ) is such that  d  dλ log w(λ) = w′(λ)  w(λ) = f (λ)  g(λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  with f and g polynomials of degree at most one and two, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Up to fractional linear transformations λ �→ (a11λ + a12)/(a21λ + a22) with [aij] ∈ GL2(R),  there are precisely four distinct weights satisfying the criterion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8):  w(λ) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  e−λ2,  Gaussian,  λae−λχλ>0,  Laguerre,  λa(1 − λ)bχ0<λ<1,  Jacobi,  (1 − iλ)η(1 + iλ)η,  generalised Cauchy,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  where the indicator function χA equals one when A is true and zero otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' its presence  is due to the fact that the Laguerre random matrix W = G†G is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking  a, b as prescribed in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 relates the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) with the Gaussian,  Laguerre, and Jacobi weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) to the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More generally, one usually  considers a, b > −1 real and η = −κ(N − 1) − 1 − α with α ∈ C and Re(α) > −1/2 for  convergence issues [39], though other values can be considered through analytic continuation  — in this general setting, the function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) continues to be referred to as an eigenvalue  15  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' even if the variables {λi}N  i=1 it governs can no longer be interpreted as eigenvalues  of an actual random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We will see in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 that there is a further generalisation  of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) where β is taken to be positive real parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like with the circular  ensembles’ eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8), we will suppress the β argument of the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) when its value is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, we will opt to use superscripts  (G), (L), (J), (Cy) to distinguish the four cases given in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The eigenvalue densities corresponding to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 are  given by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Existing derivations of their explicit forms for finite N in terms  of (skew-)orthogonal polynomials are presented in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For now, we review the large N  limiting forms of the corresponding global scaled eigenvalue densities ˜ρ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling that κ = β/2, take cN =  √  κN, κN, 1 in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) for the  Gaussian, Laguerre, and Jacobi cases, respectively [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, define the global scaled  eigenvalue densities of the classical matrix ensembles to be  ˜ρ(G)(λ) :=  �  κ  N ρ(G)(  √  κNλ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  ˜ρ(L)(λ) := κ ρ(L)(κNλ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  ˜ρ(J)(λ) := 1  N ρ(J)(λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  ˜ρ(Cy)(λ) := 1  N ρ(Cy)(λ)  ���  η=−κ(N+ˆαN−1)−1,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  with ˆα constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  There is some freedom in choosing the scaling cN, but the required characteristic of  the densities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) is that their large N limiting forms have compact connected  support on which they integrate to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Jacobi ensembles’ eigenvalue densities already  have compact connected support [0, 1], so to obtain an appropriate expression for ˜ρ(J)(λ),  one need only renormalise ρ(J)(λ) so that it integrates to one on said support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other  hand, scaling the eigenvalue densities of the (generalised) Cauchy ensembles according to  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) cannot produce a density with compact support as there is no choice of  parameter cN that leads to fast enough decay as N, |λ| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, one needs to take  η = −κ(N − 1) − 1 − α and change variables to relate to the circular Jacobi ensemble, at  which point it can be seen that when α ∝ N, one obtains a compact support in the large  16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The details of this phenomenon and some other properties that set the Cauchy  ensembles apart from the other classical ensembles are discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall that the N → ∞ limit of ˜ρ(λ) is equal to ρ0(λ), the leading order term of  the smoothed eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Gaussian case, it is given by the celebrated Wigner semi-circle  law [315],  ρ(G),0(λ) =  √  2 − λ2  π  χ|λ|<  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  If we suppose that γi := limN→∞ Mi/N exists for i = 1, 2, then we have ρ(L),0(λ) given by the  Marˇcenko-Pastur law [234],  ρ(L),0(λ) = (1 − γ1)δ(λ)χγ1<1 +  �  (λ(MP)  +  − λ)(λ − λ(MP)  −  )  2πλ  χλ(MP)  −  <λ<λ(MP)  +  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  and ρ(J),0(λ) given by the so-called Wachter law [307],  ρ(J),0(λ) = (γ1 + γ2)  �  (λ(Wac)  +  − λ)(λ − λ(Wac)  −  )  2πλ(1 − λ)  χλ(Wac)  −  <λ<λ(Wac)  +  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  with  λ(MP)  ±  := (1 ± √γ1)2 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  λ(Wac)  ±  :=  ��  γ1  γ1 + γ2  �  1 −  1  γ1 + γ2  �  ±  �  1  γ1 + γ2  �  1 −  γ1  γ1 + γ2  ��2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  Writing ˆα = ˆα1 + iˆα2 with ˆα1, ˆα2 real and constant in N, one surmises from the equivalent result for  the circular Jacobi ensemble [49, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)] that in the Cauchy case,  ρ(Cy),0(λ) = ˆα1  �  (λ(Cy)  +  − λ)(λ − λ(Cy)  −  )  π(1 + λ2)  χλ(Cy)  −  <λ<λ(Cy)  + ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  where  λ(Cy)  ±  :=  −ˆα2(ˆα1 + 1) ±  �  (ˆα2  1 + ˆα2  2)(2ˆα1 + 1)  ˆα2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Due to how κ is involved in the scalings of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6, all of the large N limiting  forms given in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 are independent of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is known that the same does not hold  for the 1/N correction terms ρ1(λ), as will be seen in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  17  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  The limiting densities ρ(G),0(λ), ρ(L),0(λ), and ρ(J),0(λ) take on parameter-independent  forms when a, b are constant in N since then, γ1 = γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They display, respectively, two  soft edges, a soft and hard edge, and two hard edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, from a topological viewpoint,  all possible combinations of soft and hard edges on a connected interval are represented by  the eigenvalue densities of the Gaussian, Laguerre, and Jacobi ensembles in the relatively  simple regime a, b = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Here, a soft edge refers to an endpoint λ of supp ρ0 with the  property that, for all large N, cNλ lies within the interior of supp ρ, while a hard edge refers to  such an endpoint when cNλ is also an endpoint of supp ρ [139], [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the present setting  concerning classical matrix ensembles, ρ0(λ) exhibits square root profiles at its soft edges  owing to tails extending past said edges being scaled to negligibility, whereas its hard edges  are located at inverse equare root singularities corresponding to strict constraints enforced  by the indicator functions seen in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1: ρ(G),0(λ)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2: ρ(L),0(λ)|γ1=1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3: ρ(J),0(λ)|γ1=γ2=1  The other regime of interest is when the parameters a and/or b are proportional to N [33],  [56], [305], [266], [229], [240], [241], [70], [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This situation is a little more complicated, as  the limiting forms ρ(L),0(λ) and ρ(J),0(λ) are no longer parameter-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, when  a (b) is linear in N, we have γ1 > 1 (γ2 > 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' consequently, the endpoint λ− (λ+) turns from  a hard edge to a soft edge, a phenomenon which is discussed further in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Cauchy  case, one sees that ρ(Cy),0(λ) also has soft edges when we take α = ˆακN with ˆα = O(1), but  unlike the ρ0(λ) of the other classical matrix ensembles, this is not a degeneration of a hard  edge: When α is constant in N, the Cauchy ensembles technically do not have a global scaled  eigenvalue density, but the large N limiting form of its equivalent is  lim  N→∞ ˜ρ(Cy)(λ)  ���  ˆα=0 = ρ(Cy),0(λ)  ���  ˆα1=ˆα2=0 =  1  π(1 + λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  As this is supported on the real line, we do not see a hard nor soft edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, one sees a  spectrum singularity at infinity, which is a term we formally introduce in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  2  1  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  2  3  4  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  Another reason for focusing on the Gaussian, Laguerre, and Jacobi ensembles while  relegating the Cauchy ensembles to secondary interest is that when considering fractional  linear transformations taken from GL2(C) rather than GL2(R), the Cauchy weight can be  seen to be equivalent to the Jacobi weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, a simple change of variables shows that  w(Cy)(λ) = 4Re(η) w(J) � 1  2(1 − iλ)  � ���  a=b=η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  Thus, our results for the Jacobi ensembles translate to the corresponding Cauchy ensembles  through analytic continuation in the parameters a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this vein, the proofs of Chapter 2  concerning the Jacobi ensembles also transfer to the Gaussian and Laguerre ensembles via  Lebesgue’s dominated convergence theorem combined with the following limiting procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let w(G)(λ), w(L)(λ), and w(J)(λ) be given by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) in the Gaussian,  Laguerre, and Jacobi cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  lim  b→∞ ba w(J)(λ/b) = w(L)(λ)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  and if we set a = b = L,  lim  L→∞ 42L w(J) �  1  2(1 + λ/  √  L)  �  = w(G)(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  Moreover, it can be observed that w(L)(λ2)|a=0 = w(G)(λ), so there is a hierarchy placing  the Jacobi ensembles above the Laguerre ensembles, which, in turn, sit above the Gaussian  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This corresponds to the loss of parameter b and then a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Some of the tools used in the proofs of Chapter 2 apply to the more general classical β  ensembles, so we defer their introductions to §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Other properties of the classical matrix  ensembles that we wish to review follow from their status as (skew-)orthogonal polynomial  ensembles (SOPEs), which are themselves examples of invariant matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These classes  of random matrix ensembles are discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Before doing so,  let us first look at the Cauchy ensembles in more depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  The Cauchy and circular Jacobi ensembles  For specific values of a, b and with β ∈ {1, 2, 4}, we have seen from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 that p(G),  p(L), and p(J) are j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the eigenvalues of random matrices that are constructed from  19  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Ginibre matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In contrast, for fixed β ∈ {1, 2, 4}, we can interpret p(Cy) as the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of a random matrix related to an appropriate circular ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To see this, let U be  drawn from the N × N circular orthogonal, unitary, or symplectic ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, define  the Hermitian random matrix A via the Cayley transformation  A := iU + IN  U − IN  = i(U + IN)(U − IN)−1 ∈ MN×N(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  The eigenvalues {λi}N  i=1 of A are related to the eigenvalues {eiθi}N  i=1 of U by the stereographic  projection  λi = cot(θi/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  Applying the associated inverse mapping eiθi = (λi + i)/(λi − i) to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) of the circular ensembles shows that the eigenvalues of A are distributed according to  the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) with the Cauchy weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), when η = −κ(N − 1) − 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  when α = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When studying the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7), there is no reason to restrict our parameters  to only those that relate the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' to a realisable random matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, for most of this  thesis, we consider the Laguerre and Jacobi ensembles with a, b > −1 real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The natural  way to generalise the Cauchy ensembles’ eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s is to introduce the complex  parameter α with Re(α) > −1/2, and write η = −κ(N − 1) − 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This ensures that  w(Cy)(λ) continues to be real-valued on the real line and that  N (Cy)  N,β =  �  RN p(Cy)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) dλ1 · · · dλN  is finite (we require Re(α) > −1/2 for finiteness due to the non-compact domain of integra-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying the stereographic projection (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) to p(Cy) with α ̸= 0 yields the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the circular Jacobi ensemble, which is an extension of the circular ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking β > 0 real, let α ∈ C, α1 := Re(α) > −1/2, and α2 := Im(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The size-  N circular Jacobi ensemble is the system of ‘eigenvalues’3 on the unit circle {eiθ | θ ∈ [0, 2π)}  distributed according to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(cJ)(eiθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , eiθN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) :=  1  N (cJ)  N,β  N  ∏  i=1  w(cJ)(eiθi) |∆N(eiθ)|β,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  3Although calling p(cJ) an eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is a priori artifical in some sense, we will see in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 that  random matrices have been constructed whose eigenvalues are indeed distributed according to p(cJ) [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar  matrix constructions for the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) with β > 0 general real are also outlined in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  where N (cJ)  N,β is a normalisation constant, ∆N(eiθ) is the Vandermonde determinant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9),  and the circular Jacobi weight [49], [226] is given by  w(cJ)(eiθ) := (1 − e−iθ)α(1 − eiθ)α = eα2(θ−π) |1 − eiθ|2α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  The weight w(cJ) is said to have a spectrum singularity of Fisher-Hartwig type [113], [30],  [325] at θ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' when α1 ̸= 0, it is of root-type (it is actually log w(cJ)(eiθ) that is singular) and  when α2 ̸= 0, it is of jump-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Accordingly, we say that the Cauchy ensembles display a  spectrum singularity in the regime |λ| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, studying the Cauchy ensembles allows us  to shine light on aspects of the spectrum singularity scaling regime, supplementing similar  studies of the soft and hard edges exhibited by the other classical ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Earlier, we  mentioned that the spectrum singularity of the Cauchy ensembles transforms into a pair of  soft edges when we take α ∝ N — the same behaviour has been recorded for the circular  Jacobi ensemble [226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This mechanism is interesting to us because when we have a spectrum  singularity, our methods involving the resolvent W1(x) cannot be used since not enough  spectral moments m(Cy)  k  converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, this ceases to be an issue when we set α = ˆακN  and work with ˜ρ(Cy)(λ) as defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be precise, note that when λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN → ∞,  |∆N(λ)|β ∼  N  ∏  i=1  λβ(N−1)  i  ,  w(Cy)(λi)|η=−κ(N−1)−1−α ∼ λ−β(N−1)−2−2α1  i  ,  so by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15), m(Cy)  k  is convergent only when −1 < k < 2α1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This upper bound  disappears upon setting α1 = ˆα1κN, as introduced in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, and taking N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In this thesis, the connection between the Cauchy and circular Jacobi ensembles will not be  used to study the Cauchy ensembles, but will rather serve as motivation for doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead,  as mentioned earlier, our results concerning the Cauchy ensembles flow from equivalent  results for the Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Along the way, we will also formulate certain intermediate  results in terms of the shifted Jacobi ensembles, which are defined to have eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) with weight  w(sJ)(λ) := (1 − λ)a(1 + λ)bχ−1<λ<1 = w(J) � 1  2(1 − λ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  Since results for the Cauchy and shifted Jacobi ensembles descend from those for the (un-  shifted) Jacobi ensembles, we present only those results where the relevant transformation  gives rise to significant simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, we will report primarily on the symmetric  Cauchy and shifted Jacobi ensembles, which corresponds to taking α ∈ R and a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  21  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Invariant matrix ensembles  Given a group G and a group action A : G × S → S, a random matrix ensemble E = (S, P)  is said to be an A-invariant matrix ensemble if the probability measure dP(X) := P(X) dX  (recall that in the case of the circular ensembles, we do not prescribe a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X), but work  with a probability measure dP(X) directly) satisfies  dP(A(g, X)) = dP(X)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  for every g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, the CUE of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 is invariant under left and right  multiplication by elements of U(N), while the COE and CSE have invariant structures  induced by those of the CUE [98, I, Thrms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The adjoint action is by far the most prolific group action studied in random matrix  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, the term invariant matrix ensemble, with A unspecified, refers to invariance  under the mapping X �→ gXg−1 for all g drawn from some appropriate compact group G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  when such an invariant matrix ensemble is assumed to be irreducible and the matrix set  underlying it is such that S ⊆ MN×N(R), MN×N(C), or MN×N(H), the group G must be  one of O(N), U(N), or Sp(2N) [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the invariant matrix ensemble at hand is more  precisely referred to as an orthogonal (O(N)), unitary (U(N)), or symplectic (Sp(2N)) ensemble,  and if the generic entries of the elements of both S, G belong to the same number system, the  statistics of the ensemble are basis-independent (conjugation by an element of G is equivalent  to a change of basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The adjectives orthogonal, unitary, and symplectic have already been  introduced in the circular and Ginibre cases in Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, where we saw that  they correspond to β = 1, 2, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It can moreover be checked that these  invariances extend to the Cauchy matrix A defined in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) and we show below  that the real, complex, and quaternionic matrices of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 have orthogonal, unitary,  and symplectic invariance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, with G, L, J, Cy labelling the classical weights  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and O, U, S the invariance classes, we will for the remainder of this thesis refer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  to the real Gaussian ensemble as the GOE and the β = 4 Cauchy ensemble as the CySE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Invariance of the circular, Ginibre, Gaussian, Laguerre, and Jacobi ensembles  In the CUE case, dP(X) is the Haar measure on U(N), so it is trivially invariant under  the mapping X �→ UXU† for U ∈ U(N) — it is a slightly more astute observation that  22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  this invariance translates to invariance of the COE (CSE) under the adjoint action of O(N)  (Sp(2N)) [98, I] (recall the forms of S given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Otherwise, the probability  measure dP(X) can be seen to be invariant under the adjoint action of a group G if one is able  to establish the invariance of the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) and the Lebesgue measure dX separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is  immediate that P(Gin)|M=N(X), P(G)(X), P(L)(X), and P(J)(X), as specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, are invariant under the appropriate adjoint actions (conjugation of X  by elements of O(N), U(N), Sp(2N) corresponding to β = 1, 2, 4) since the functions Det X,  Tr X†X, and Tr Xk (k ∈ N) all have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More involved is the invariance of the  Lebesgue measure dX, where the strategy used depends on whether or not the associated  matrix set S consists purely of self-adjoint matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In regards to Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, one may use  the diagonalisation formula X = UΛU† and some calculus of differential forms to see that  the Lebesgue measure on the sets of self-adjoint matrices S(G), S(L), S(J) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) can  be written in a form that is manifestly invariant under the relevant adjoint action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Given an N × N real (β = 1), complex (β = 2), or quaternionic (β = 4)  self-adjoint matrix variable X, diagonalise it as X = UΛU†, where Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) is a  diagonal matrix of eigenvalues, and U is a corresponding matrix of eigenvectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the spectral theorem  dictates that U is an element of O(N) in the real case, U(N) in the complex case, or Sp(2N) in the  quaternionic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To make the mapping X �→ UΛU† bijective, we insist that the eigenvalues be listed  in increasing order and that the first row of U be non-negative real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The canonical Lebesgue measure  dX =  N  ∏  i=1  dXii  ∏  1⩽j<k⩽N  β  ∏  s=1  dX(s)  jk  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  on the set of self-adjoint matrices (recall that X(s)  jk denotes the sth real component of the matrix entry  Xjk, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the measures (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)) decomposes as  dX = |∆N(λ)|β dΛ dµHaar(U),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  where dΛ = ∏N  i=1 dλi is the Lebesgue measure on the Weyl chamber {λ1 < · · · < λN} ⊂ RN and  dµHaar(U) =  �  1⩽l<k⩽N  β�  s=1  �  U† �  dUij  �N  i,j=1  �(s)  lk  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  is the (unnormalised) Haar measure on the quotient space O(N)/O(1)N (β = 1), U(N)/U(1)N  (β = 2), or Sp(2N)/Sp(2)N (β = 4) given by the exterior (wedge) product of the independent real  components of the entries of the matrix of differentials U† �  dUij  �N  i,j=1 [180], [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  23  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using Leibniz’s rule, take the exterior derivative of the diagonalisation formula  X = UΛU† and then conjugate the result by U† to obtain  U† �  dXij  �N  i,j=1 U = U† �  dUij  �N  i,j=1 Λ − Λ U† �  dUij  �N  i,j=1 + diag(dλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , dλN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  Carrying out the necessary matrix multiplications shows that the right-hand side is a self-  adjoint matrix with the (i, j) entry given by dλiχi=j + (λj − λi)  �  U†[dUkl]N  k,l=1  �  ij for i ⩽ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Taking the exterior product of the real components of these entries and replacing the Jacobian  by its absolute value yields the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), while applying the same  prescription to the entries of the left-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) gives dX, as desired  — note that  �  U†[dUkl]N  k,l=1  �  ij contains β real components, resulting in as many powers of  |∆N(λ)| in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The quotient spaces O(N)/O(1)N, U(N)/U(1)N, Sp(2N)/Sp(2)N are  respectively isomorphic to the subsets of O(N), U(N), Sp(2N) that house the matrix U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Haar measures on O(N), U(N), Sp(2N) are computed from U†[dUij]N  i,j=1 in the  same way as dµHaar(U) above, but the outcome is different due to the extra N parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Indeed, if U ∈ O(N), U(N), Sp(2N) instead of O(N)/O(1)N, U(N)/U(1)N, Sp(2N)/Sp(2)N,  the relevant procedure returns  dµ′  Haar(U) =  N  �  h=1  �  U† �  dUij  �N  i,j=1  �  hh  �  1⩽l<k⩽N  β�  s=1  �  U† �  dUij  �N  i,j=1  �(s)  lk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  This is not to be confused with the measures defining the circular ensembles, where equiva-  lence holds only in the CUE case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It is now straightforward to see that the Lebesgue measure on the set of self-adjoint  matrices is invariant under the adjoint action: Taking the complex case for definiteness, write  X ∈ S(G) as X = UΛU† according to the prescription given in Propositon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, let U′ ∈ U(N),  and define X′ = U′XU′†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, letting D be a diagonal matrix of phases such that U′UD  has first row non-negative real, we have X′ = (U′UD)Λ(U′UD)† and hence  dX′ = |∆N(λ)|2 dΛ dµHaar(U′UD) = dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  The second equality follows from (U′UD)† �  d(U′UD)ij  �N  i,j=1 = D†U† �  dUij  �N  i,j=1 D and the  fact that the exterior product of the independent real components of the entries of this second  matrix agrees with dµHaar(U) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33), up to sign [235, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20], [119, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This latter  24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  equivalence of measures can be checked by writing D = ∏N  l=1 Dl with Dl := [Dijχi=j=l]N  i,j=1  and explicitly showing for a matrix of differentials Ω = [dωij]N  i,j=1 with only upper triangular  entries independent (as is true of U† �  dUij  �N  i,j=1) that  �  1⩽j<k⩽N  (DlΩD†  l )(1)  jk ∧ (DlΩD†  l )(2)  jk =  �  1⩽j<k⩽N  dω(1)  jk ∧ dω(2)  jk ,  1 ⩽ l ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  Extending these arguments to the real and quaternionic cases verifies that the Gaussian,  Laguerre, and Jacobi ensembles, as introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 and further specified by  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, are invariant matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar to the self-adjoint matrices above, elements X of the circular ensembles  have unique eigendecompositions of the form X = U exp(iΘ)U†, where U is as given in  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and Θ = diag(θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , θN) is such that the eigenangles {θi}N  i=1 are listed in  increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The analogue of the decomposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) in the circular cases is  dP(X) =  1  Z(C)  N,β  |∆N(eiθ)|β dΘ dµHaar(U),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  where dµHaar(U) is again as defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, dΘ = ∏N  i=1 dθi is the Lebesgue  measure on {0 ⩽ θ1 < · · · < θN < 2π} [98, I], and Z(C)  N,β is a normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note  that in both equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38), the Jacobian drawn out from the change of  variables is the absolute value of the Vandermonde determinant to the βth power, making  the related invariant matrix ensembles suitable for modelling systems of random particles  that interact via pairwise logarithmic repulsion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', log-gases [119]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the situation concerning N × N Ginibre matrices G, things are more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the 1965 work [158], Ginibre extended the strategy outlined above by starting with  the diagonalisation formula G = PΛP−1, where Λ is a matrix of eigenvalues and P is a  corresponding matrix of eigenvectors (this formula is valid with probability one since the  subset of matrices with repeated eigenvalues has Lebesgue measure zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since G is not  self-adjoint, its eigenvalues are generically complex in the complex and quaternionic cases  (when G is quaternionic, it has N pairs of complex conjugate eigenvalues that are thus fully  determined by those in the upper half plane), while in the real case, G has a mixture of real  and complex conjugate eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The eigenvalues of G do not have a natural ordering,  but this can be accounted for by a factor of N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' or by assigning some arbitrary ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More  25  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  importantly, the diagonalising matrix P is not necessarily orthogonal, unitary, nor symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Nonetheless, the QR decomposition can be used to write P = UT where U is as specified in  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, and T is an upper triangular matrix with diagonal entries real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, it was shown in [158] that the relevant Lebesgue measure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) decomposes as  dG = J(λ) dΛ dµTri(T) dµHaar(U),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  where dΛ is the Lebesgue measure on the space housing the eigenvalues, dµHaar(U) is  the Haar measure defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, J(λ) is a Jacobian consisting of products  of Vandermonde determinants involving the eigenvalues, and dµTri(T) is some measure  dependent on the entries of T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we leave the latter two unspecified and refer the interested  reader to [158].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As was argued for the decomposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), it follows from the factorisation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) that dG is invariant under the desired adjoint action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Combined with the invariance  of P(Gin)|M=N(G) established earlier, we conclude that the N × N Ginibre ensembles are  invariant matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Soon after, Dyson [238, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33] provided a simplification of the above in the com-  plex and quaternionic cases by using the Schur decomposition G = U ˆTU†, with U as in  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and ˆT upper triangular with diagonal entries being the eigenvalues of G —  the measure dG was yet again shown to factorise in a similar manner to that in equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that one can choose ˆT = TΛT−1, where T is the upper triangular matrix used  by Ginibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The real case was treated in this manner by Edelman in 1997 [100] in a parallel  effort to Lehmann and Sommers’ 1991 work [225], with ˆT now upper quasi-triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  quasi-triangular form is due to the complex conjugate eigenvalue pairs being encoded within  appropriate 2 × 2 real matrices, which is necessitated by the requirement that ˆT be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Relation to eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s  To obtain the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) from the probability measure dP(X), one  changes variables to the eigenvalues and a set of auxiliary variables, then integrates over  the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the ensembles discussed above, the auxiliary variables relate to the entries  of the matrix U ∈ O(N)/O(1)N (β = 1), U(N)/U(1)N (β = 2), Sp(2N)/Sp(2)N (β = 4)  and, in the Ginibre cases, either the entries of T or the strictly upper triangular entries of  ˆT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Ginibre [158] was able to integrate out the auxiliary variables and obtain the eigenvalue  26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for the complex and quaternionic Ginibre ensembles, but could only address the  real Ginibre ensemble in the case that all of the eigenvalues were real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Approximately  25 years later, the works [225], [100] used (a form of) the Schur decomposition to obtain  the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the real Ginibre ensemble with any number of real eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Although Ginibre’s substitution G = UTΛT−1U† and the Schur decomposition G = U ˆTU†  both lead to expressions for dG that factorise nicely, only the latter transforms P(Gin)|M=N(G)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) into an amicable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, the benefit of using the Schur decomposition is that  P(Gin)|M=N(U ˆTU†) = ˆP( ˆT)  N  ∏  i=1  e−λ2  i ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  where ˆP( ˆT) is some function depending purely on the strictly upper triangular entries of ˆT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the interest of brevity, we do not review the Ginibre ensembles in any further detail  here, but note that their eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s have vastly different forms for β = 1, 2, 4, in  contrast to the universal structures of the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) of the circular  and classical matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A succinct review of the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the Ginibre  ensembles can be found in [128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Contrary to the eigenvalues, the squared singular values of the Ginibre matrix G  are non-negative real and are well-defined even when G is rectangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, they are  equivalent to the eigenvalues of the Wishart–Laguerre matrix G†G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moving on to the Gaussian, Laguerre, and Jacobi ensembles, things are simpler than in  the Ginibre cases since our diagonalising matrices are now orthogonal, unitary, or symplectic,  as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Combined with the invariance of the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X), this means that under the  change of variables X = UΛU† prescribed by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we have that P(X) = P(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Hence, letting P(X) be one of the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s specified in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and fixing (β, G(N)) =  (1, O(N)), (2, U(N)), or (4, Sp(2N)), Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 implies that  �  G(N)/G(1)N dP(X) = P(Λ) |∆N(λ)|β dΛ  �  G(N)/G(1)N dµHaar(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  As the eigenvalue and eigenvector statistics are decoupled, one reads off that the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is thus given by  p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) = vol(G(N)/G(1)N)  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(Λ) |∆N(λ)|β,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  27  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  where  vol(G(N)/G(1)N) :=  �  G(N)/G(1)N dµHaar(U)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43)  is the volume of G(N)/G(1)N with respect to the Haar measure defined in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3,  while the factor of N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is needed to compensate our removal of the ordering on the eigenvalues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  the ordering λ1 < · · · < λN is implicit in the definition of dΛ, but clashes with the definition  of p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN), which assumes that all eigenvalues are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  With w(λ)  the appropriate classical weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), it is easy to check that P(Λ) = ∏N  i=1 w(λi), which  confirms that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) is consistent with the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) of the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, we are led to the observation that the partition functions of  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 can be obtained from the computation  ZN = vol(G(N)/G(1)N)  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  NN,β,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  as long as the value of vol(G(N)/G(1)N) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This volume can be computed in a  few ways, one of which being a geometric argument of Dyson’s given in [98, I], and in  fact already known and implemented by Hurwitz [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another approach is to simply  compute the ratio N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='ZN/NN,β in a tractable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, since P(G)(X) is a product of  independent Gaussian p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s, it is straightforward to see that Z(G)  N,β = (π)N/2(π/2)N(N−1)β/4,  while N (G)  N,β can be derived from the Selberg integral theory outlined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 or simply read  off from [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In short, recalling the (unnormalised) Haar measure dµ′  Haar(U) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35),  vol(G(N)/G(1)N) =  �  G(N) dµ′  Haar(U)  ��  G(1) dµ′  Haar(U)  �N = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='πN(N−1)β/4  N  ∏  j=1  Γ(β/2 + 1)  Γ(βj/2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45)  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The above argument translates directly to the circular ensembles, with Λ replaced  by exp(iΘ) and dP(X) specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The analogue of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) is  then  p(C)(eiθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , eiθN) = vol(G(N)/G(1)N)  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  |∆N(eiθ)|β,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  which is consistent with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling the values of N (C)  N,β given in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1,  we see that  Z(C)  N,β = vol(G(N)/G(1)N)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  exactly when β = 2 and G(N) = U(N), in keeping with Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  Other invariant matrix ensembles  When establishing the invariance of the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P(X) of the Gaussian, Laguerre, and Jacobi  ensembles given in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we used the fact that they are functions of Det X and  Tr Xk for k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This argument extends naturally to the situation where the parameters a, b  are continuous, even though their origin in terms of (rectangular) Ginibre matrices restricts  them to a discrete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, it makes sense to endow the matrix sets S(L), S(J) with the  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P(L)(W), P(J)(Y) with general real parameters a, b > −1, seeing as then the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' induced by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) is integrable and supported on an interval in the real  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, we are able to give the Cauchy analogue of these p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let β = 1, 2, 4, equivalently κ = 1/2, 1, 2, correspond to fixing the field F as  R, C, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, matrices X drawn from S(G) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), the set of self-adjoint N × N matrices, with  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(Cy)(X) :=  1  Z(Cy)  N,β,α  Det(IN + X2)−κ(N−1)−1−α  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48)  have eigenvalues distributed according the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(Cy) as defined by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) with  Cauchy weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The partition function Z(Cy)  N,β,α is given by N (Cy)  N,β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) multiplied by the  appropriate volume (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) as prescribed by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, there exist invariant matrix ensembles that correspond to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  for every classical weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and all valid parameters a, b, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' All that is lost due to this  formulation is that the relevant random matrices can no longer be related to the Ginibre  ensembles in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rather, we will see in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 that there are constructions involving  tridiagonal matrices which hold for continuous parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42), one is able to interpret a given multivariable symmetric  function p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) as the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of a self-adjoint real, complex, or quaternionic  random matrix X = UΛU†, with Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) and U ∈ O(N), U(N), Sp(2N) Haar-  distributed according to dµHaar(U) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, it is known from symmetric function  theory [232] that the factor P(Λ) in the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) is a function of  {Tr Λk}N  k=0, hence {Tr Xk}N  k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the matrix X has a well-defined p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) on a subset  of the space of self-adjoint matrices, assuming that p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) is sufficiently integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  29  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Since the upcoming (skew-)orthogonal polynomial ensembles and classical β ensembles  are specified by eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the type considered in the above remark, they can be  interpreted as invariant matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As we end this subsection, let us remark that the  matrix product ensembles discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 are also invariant matrix ensembles due to the  invariance of the Ginibre and Laguerre ensembles (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [185] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Skew-orthogonal polynomial ensembles  The subclass of invariant matrix ensembles considered in this subsection are those concerning  self-adjoint matrices X with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P(X) such that one has the decomposition  P(X) =  N  ∏  i=1  w(λi),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49)  where {λi}N  i=1 are the indistinguishable eigenvalues of X and w(λ) is some continuous  weight function that is real-valued on R and decays fast enough as |λ| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equivalently,  we are interested in j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7), but with the weights no longer constrained  to be classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let β = 1, 2, or 4, fix N ∈ N, and let w(λ) be a continuous, non-negative,  real-valued function such that w(λ) = o(λ−β(N−1)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (w)  N,β  N  ∏  i=1  w(λi) |∆N(λ)|β  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50)  describes a size-N orthogonal polynomial ensemble (OPE) when β = 2, skew-orthogonal polynomial  ensemble of real type (R-SOPE) when β = 1, or skew-orthogonal polynomial ensemble of quaternion  type (H-SOPE) when β = 4 [238, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As usual, N (w)  N,β is a normalisation constant and  ∆N(λ) is the Vandermonde determinant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The condition w(λ) = o(λ−β(N−1)−1) ensures that the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' specified by the above  definition has a well-defined (convergent) normalisation constant, but it does not necessarily  have convergent spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As an example, the Cauchy ensembles of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 exhibit  these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To understand the nomenclature introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8, a few key observations  regarding the Vandermonde determinant need to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following [238, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5], [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 6], let {qj(λ)}∞  j=0 be a set of monic polynomials such  that qj(λ) is degree j for each j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  ∆N(λ) := Det  �  λj−1  i  �N  i,j=1 = Det  �  qj−1(λi)  �N  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)  ∆N(λ)4 = Det  �  �  λj−1  i  (j − 1)λj−2  i  �  �  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',N  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',2N  = Det  �  �qj−1(λi)  q′  j−1(λi)  �  �  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',N  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',2N  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  and when N is even (N odd can be treated by setting sgn(λN+1 − λi) = 1 in the Pfaffian factor),  |∆N(λ)| = ∆N(λ) Pf  �  sgn(λj − λi)  �N  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53)  where sgn(0) = 0 and otherwise, sgn(λ) = λ/|λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) from the definition of the  Vandermonde determinant presented on its immediate left, one simply notes that the  jth column of the right-hand side is equal to the jth column of the Vandermonde matrix  [λj−1  i  ]N  i,j=1 plus columns to its left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since adding columns does not change the determinant,  one can replace the the columns [λj−1  i  ]N  i=1 with [qj−1(λi)]N  i=1 by successively iterating through  j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The middle expression of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) can be obtained through induction  on N, and the right-hand side follows from a column-operation argument similar to that for  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) follows from the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and  the identity [76] (valid for even N, see Appendix A)  ∏  1⩽i<j⩽N  sgn(λj − λi) = Pf  �  sgn(λj − λi)  �N  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  which can be checked by confirming equality in the case λ1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' < λN and comparing sign  changes upon interchanging variables λi ↔ λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  If one wishes to integrate the determinantal and Pfaffian structures given in the above  lemma against the product ∏N  i=1 w(λi) occurring in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50), it is beneficial to use  monic polynomials {qj(λ)}∞  j=0 that are (skew-)orthogonal with respect to the weight w(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With w(λ) a suitable weight function (usually taken to have all integer  moments finite, but satisfying the hypotheses given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 is sufficient for our  31  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  purposes), define the following inner products on polynomial space [238, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5], [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 6]:  ⟨ f, g⟩(1) := 1  2  �  R2 f (x)g(y)sgn(y − x)w(x)w(y) dxdy,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='55)  ⟨ f, g⟩(2) :=  �  R f (x)g(x)w(x) dx,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56)  ⟨ f, g⟩(4) := 1  2  �  R  �  f (x)g′(x) − f ′(x)g(x)  �  w(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57)  For β = 1, 2, 4, let {q(β)  j  (λ)}∞  j=0 be a family of monic polynomials such that each q(β)  j  (λ) is  degree j, and further posit that there exist finite, positive constants {r(β)  j  }∞  j=0 such that  �  q(2)  j  , q(2)  k  �(2)  = r(2)  j  χj=k,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58)  �  q(σ)  2j , q(σ)  2k+1  �(σ)  = −  �  q(σ)  2k+1, q(σ)  2j  �(σ)  = r(σ)  j  χj=k,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='59)  �  q(σ)  2j , q(σ)  2k  �(σ)  =  �  q(σ)  2j+1, q(σ)  2k+1  �(σ)  = 0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='60)  for all j, k ⩾ 0 and σ = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, we say that the family {q(1)  j (λ)}∞  j=0 is R-skew-orthogonal,  {q(2)  j (λ)}∞  j=0 is orthogonal, and {q(4)  j (λ)}∞  j=0 is H-skew-orthogonal, all with respect to w(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The existence of these polynomials is assured by a Grahm–Schmidt type construc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They are unique for β = 2, but for β = 1, 4, the skew-orthogonality remains true upon  the replacement of q(β)  2j+1(λ) by q(β)  2j+1(λ) + ξ(β)  2j q(β)  2j (λ) for arbitrary ξ(β)  2j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Utilising these (skew-)orthogonal polynomials in the expressions given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, it is  possible to express the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50) as either a determinant  or a Pfaffian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We present this result without reviewing its derivation, but note that the  essential idea is to absorb the factors of w(λi) into the structures presented in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, [238]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following [238, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5], let {q(β)  j  (x)}∞  j=0, {r(β)  j  }∞  j=0 be as in  the preceding definition, and let N ∈ N with N even in the case β = 1 (N odd is not much more  complicated, but will not be covered here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Next, introduce the auxiliary functions  ψj(x) :=  �  R q(1)  j (y)sgn(x − y)w(y) dy,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61)  S(1)  j (x, y) := ψ′  2j+1(x)ψ2j(y) − ψ′  2j(x)ψ2j+1(y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='62)  S(4)  j (x, y) := q′  2j+1(x)q2j(y) − q′  2j(x)q2j+1(y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63)  D(4)  j  (x, y) := q′  2j+1(x)q′  2j(y) − q′  2j(x)q′  2j+1(y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='64)  I(4)  j  (x, y) := q2j+1(x)q2j(y) − q2j(x)q2j+1(y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65)  32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  and the kernels corresponding respectively to the real, complex, and quaternionic cases,  K(1)  N (x, y) :=  N/2−1  ∑  k=0  1  r(1)  k  �  �  1  2  �  R sgn(x − z)S(1)  k (z, y) dz  S(1)  k (y, x)  −S(1)  k (x, y)  ∂yS(1)  k (x, y)  �  �  − 1  2  �  �sgn(x − y)  0  0  0  �  � ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='66)  K(2)  N (x, y) :=  �  w(x)w(y)  N−1  ∑  k=0  1  r(2)  k  q(2)  k (x)q(2)  k (y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='67)  K(4)  N (x, y) :=  �  w(x)w(y)  N−1  ∑  k=0  1  r(4)  k  �  � I(4)  k (x, y)  S(4)  k (y, x)  −S(4)  k (x, y)  D(4)  k (x, y)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='68)  Then, we have the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 given by  p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  �  �  �  �  �  �  �  Det  �  K(2)  N (λi, λj)  �N  i,j=1 ,  β = 2,  Pf  �  K(β)  N (λi, λj)  �N  i,j=1 ,  β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='69)  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='69) can be made uniform by interpreting  K(1)  N (x, y) and K(4)  N (x, y) as complex quaternions (see Note A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 of Appendix A) and replacing  the Pfaffians by quaternionic determinants, following equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The significance of the above theorem lies in the fact that for β = 1, 2, 4, the kernels  K(β)  N (x, y) have the reproducing properties [99]  �  R Det  �  K(2)  N (λi, λj)  �k+1  i,j=1 dλk+1 = (N − k) Det  �  K(2)  N (λi, λj)  �k  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='70)  �  R Pf  �  K(σ)  N (λi, λj)  �k+1  i,j=1 dλk+1 = (N − k) Pf  �  K(σ)  N (λi, λj)  �k  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='71)  where σ = 1, 4 and 1 ⩽ k ⩽ N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, one may iteratively integrate p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN)  over dλk+1 · · · dλN (1 ⩽ k ⩽ N − 1) to obtain the k-point correlation function  ρ(w)  k  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) :=  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  �  RN−k p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) dλk+1 · · · dλN  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='72)  =  �  �  �  �  �  �  �  Det  �  K(2)  N (λi, λj)  �k  i,j=1 ,  β = 2,  Pf  �  K(β)  N (λi, λj)  �k  i,j=1 ,  β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='73)  The structures on the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='73) characterise the OPEs and SOPEs  as determinantal and Pfaffian point processes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They imply, in particular, that the  33  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  eigenvalue density ρ(w)(λ) = ρ(w)  1  (λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) is given by the remarkably elegant formula  ρ(w)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  �  �  �  �  �  K(2)  N (λ, λ),  β = 2,  Pf  �  K(β)  N (λ, λ)  �  ,  β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='74)  The right-hand sides of equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='73) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='74) can be simplified further still by  using the (generalised) Christoffel–Darboux formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Focusing first on the β = 2 case and  recalling the notation introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9, it is well known [293] that the orthogonal  polynomials {q(2)  j (λ)}∞  j=0 satisfy the three-term recurrence  q(2)  j (λ) = (λ + aj)q(2)  j−1(λ) −  r(2)  j−1  r(2)  j−2  q(2)  j−2(λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='75)  where the coefficients {aj}∞  j=1 depend on the choice of weight w(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using this recurrence,  one can show that  q(2)  j+1(x)q(2)  j (y) − q(2)  j (x)q(2)  j+1(y) = (x − y)q(2)  j (x)q(2)  j (y)  +  r(2)  j  r(2)  j−1  �  q(2)  j (x)q(2)  j−1(y) − q(2)  j−1(x)q(2)  j (y)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='76)  which can then be rearranged to give  q(2)  j (x)q(2)  j (y)  r(2)  j  =  1  x − y  �  �q(2)  j+1(x)q(2)  j (y) − q(2)  j (x)q(2)  j+1(y)  r(2)  j  −  q(2)  j (x)q(2)  j−1(y) − q(2)  j−1(x)q(2)  j (y)  r(2)  j−1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='77)  Telescopically summing this equation over j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N − 1 (with q(2)  −1 ≡ 0) reveals the  Christoffel–Darboux formula:  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The β = 2 kernel defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 simplifies as [293]  K(2)  N (x, y) =  �  w(x)w(y)  r(2)  N−1  q(2)  N (x)q(2)  N−1(y) − q(2)  N−1(x)q(2)  N (y)  x − y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='78)  Hence, the sum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='67) involving the orthogonal polynomials q(2)  0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , q(2)  N−1 can be seen  to depend only on q(2)  N and q(2)  N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking the limit y → x using L’Hˆopital’s rule shows that  for finite N, the eigenvalue density has the form  ρ(w)(λ) = K(2)  N (λ, λ) = w(λ)  r(2)  N−1  �  q(2)  N−1(λ)∂λq(2)  N (λ) − q(2)  N (λ)∂λq(2)  N−1(λ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='79)  34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  All that remains to make ρ(w)(λ) fully explicit is knowledge of the orthogonal polynomials  {q(2)  j (λ)}∞  j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Gaussian, Laguerre, and Jacobi cases, these are the Hermite, Laguerre,  and Jacobi orthogonal polynomials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Cauchy case, one requires the Routh–  Romanovski polynomials [281], which are related to the Jacobi polynomials by the change of  variables implicit in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Obtaining the β = 1, 4 analogues of equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='78) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='79) is a little more involved,  as one might expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The initial challenge was to pin down suitable skew-orthogonal  polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The works [258], [259], [4], [156] outline (progressively simpler) constructions  of skew-orthogonal polynomials for the Gaussian, Laguerre, and Jacobi weights from the  polynomials that are orthogonal with respect to these weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In these works, the correlation  kernels K(β)  N (x, y) and eigenvalue densities  ρ(w)(λ) = w(λ)  N−1  ∑  k=0  1  r(β)  k  S(β)  k (λ, λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='80)  with β = 1, 4, were not given by Christoffel–Darboux formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, ∑N−1  k=0 S(β)  k (x, y)/r(β)  k  was presented as (essentially, up to factors of two) the Christoffel–Darboux sum (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='78)  plus a ‘correction term’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Later on, [154] showed that the skew-orthogonal polynomials  satisfy recurrence relations, which were used to derive generalised Christoffel–Darboux formulae  involving the semi-infinite vectors  Φ(β)(λ) = w(λ)  �  (r(β)  2i )−1/2q(β)  2i (λ),  (r(β)  2i+1)−1/2q(β)  2i+1(λ)  �  i=0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ,  λΦ(β)(λ), ∂λΦ(β)(λ), and ∂λ[λΦ(β)(λ)], together with the semi-infinite matrices describing  the transitions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Gaussian, Laguerre, and Jacobi cases, the aforemen-  tioned recurrence relations involve three terms [155], so the resulting generalised Christoffel–  Darboux formulae are relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  Classical β ensembles  The (skew-)orthogonal polynomial ensembles of the previous subsection concern the eigen-  value j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) of the classical matrix ensembles generalised to non-classical weights  w(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this subsection, we return to w(λ) being classical, and instead allow general  real β > 0, rather than just β = 1, 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the classical β ensembles are specified by the  35  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (w)  N,β  N  ∏  i=1  w(λi) |∆N(λ)|β,  β = 2κ > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81)  with w(λ) a classical weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In addition to taking β to be positive real, we also  consider the classical weights with parameters a, b > −1 real and η = −κ(N − 1) − 1 − α,  α ∈ C with Re(α) > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' At the end of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we argued that even at this level of generality,  these eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s describe the eigenvalues of certain ensembles of self-adjoint random  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, generating matrix representatives of these ensembles is not practical  since the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P(X) of the matrices X do not translate to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s on the entries of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence,  there was a natural motivation to construct alternative matrix models for the classical β  ensembles, with a focus on specifying p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s on matrix entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 (Dumitriu–Edelman ’02, Killip–Nenciu ’04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall that random variables x ∼ χ(k)  and y ∼ B(s, t) are respectively chi and beta distributed if they have p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s  2k/2−1  Γ(k/2) xk−1e−x2/2χx>0,  21−s−tΓ(s + t)  Γ(s)Γ(t)  (1 − y)s−1(1 + y)t−1χ−1<y<1,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='82)  and let N(0, σ) denote the normal distribution with mean 0 and variance σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduce the matrices  Hβ := 1  2  �  ����������  x(G)(1)  y(G)(1)  y(G)(1)  x(G)(2)  y(G)(2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  y(G)(N − 2)  x(G)(N − 1)  y(G)(N − 1)  y(G)(N − 1)  x(G)(N)  �  ����������  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='83)  Bβ :=  1  √  2  �  �������  x(L)(1)  y(L)(1)  x(L)(2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  y(L)(N − 1)  x(L)(N)  �  �������  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='84)  and  Yβ := 1  4  �  ����������  x(J)(1)  y(J)(1)  y(J)(1)  x(J)(2)  y(J)(2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  y(J)(N − 2)  x(J)(N − 1)  y(J)(N − 1)  y(J)(N − 1)  x(J)(N)  �  ����������  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='85)  36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  with the entries independently distributed according to  x(G)(k) ∼ N(0, 2),  x(L)(k) ∼ χ (2a + 1 + β(N − k)) ,  y(G)(k), y(L)(k) ∼ χ (β(N − k)) ,  x(J)(k + 1) = (1 − α2k−1)α2k − (1 + α2k−1)α2k−2 + 2,  y(J)(k + 1) =  �  (1 − α2k−1)(1 − α2  2k)(1 + α2k+1),  where α2N−1 = α−1 = −1 and for 0 ⩽ k ⩽ 2N − 2,  αk ∼  �  �  �  �  �  B( 2N−k−2  4  β + b + 1, 2N−k−2  4  β + a + 1),  k even,  B( 2N−k−3  4  β + a + b + 2, 2N−k−1  4  β),  k odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Then, the eigenvalues of the tridiagonal real symmetric N × N matrices Hβ, Wβ := BβBT  β, and  Yβ have eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN), as specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81), with the Gaussian,  Laguerre, and Jacobi weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), respectively [93], [206].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The proofs given in [93] for the Gaussian and Laguerre cases of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 consist of  multiplying together the independent entries of the relevant matrices, changing variables  to the eigenvalues and eigenvectors, and then integrating out the eigenvector contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  These constructions are motivated by viewpoints in matrix reduction from numerical linear  algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, in the GOE and GUE, Householder transformations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [162])  can be used to systematically reduce the corresponding real, respectively complex Hermitian,  matrices to the tridiagonal form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Another matrix model for the Gaussian case [123, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3], more in line with the recursive  theme of this thesis, involves the iterative construction of the sequence of matrices  MN+1 =  �  �MN  y  yT  x  �  � ,  y = [y(i)]N  i=1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='86)  with x ∼ N(0, 2) and y(i) ∼ χ(βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Replacing MN by diag(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN), the diagonal matrix  of its eigenvalues, does not change the eigenvalue distribution of MN+1, and further allows  one to read off the recurrence  ChN+1(λ)  ChN(λ)  = λ − x −  N  ∑  i=1  y(i)2  λ − xi  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='87)  37  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  for the characteristic polynomial ChN(λ) = Det(λIN − MN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking the residues of both  sides of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='87) at λ = xi (taking x = 0 without loss of generality), one may express  the {y(i)}N  i=1 in terms of the {xi}N  i=1 and the eigenvalues {λi}N+1  i=1  of MN+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting  such expressions into the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for y and incorporating the necessary Jacobian yields a  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˆp(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) on the eigenvalues of MN+1 conditional on the eigenvalues  of MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We leave this j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' unspecified here (it is fully specified in [123]) but conclude with  the fact [123] that, after appropriate scaling, the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(G)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) defined  by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) with the Gaussian weight satisfies the recurrence  p(G)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN+1) =  �  λN+1<xN<···<λ1  p(G)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) ˆp(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) dx1 · · · dxN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  That is, the eigenvalues of MN are distributed according to the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the Gaussian β  ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The matrix Yβ given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 was derived in [206] by first considering a β-  generalisation of SO(2N) with its Haar probability measure, and then using a stereographic  projection so as to map the eigenvalues from the unit circle to an interval in the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In addition, [206] provided a matrix model for the circular β ensemble, which is specified  by the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) with β now positive real — it is a β-generalisation of the  CUE in the same vein as that considered for SO(2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Without going into details, [206] used  theory concerning orthogonal polynomials on the unit circle to construct matrices whose  eigenvalues are distributed according to the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) of the circular β ensemble with  general β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These matrices are themselves products of two block-diagonal matrices whose  entries have fully specified p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s, so they can be generated numerically in a straightforward  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This work was extended in [49] through a certain α-deformation to produce matrix  models for the circular Jacobi ensemble defined in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Through the Cayley transformation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), the matrix models for the circular Jacobi ensemble essentially serve as such for the  Cauchy β ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying Householder transformations to Ginibre matrices gives rise  to Hessenberg matrices, with the upper triangular entries still normally distributed  elements of the respective number system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, without the self-adjoint symmetry of  the GOE, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the simplicity of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='83) is lost and there is no general-β matrix model  for the Ginibre ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Classical Matrix Ensembles  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The eigenvalue densities corresponding to the classical β ensembles have the large N  limiting forms given in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 with the identifications (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  γ1 = 1 + lim  N→∞  a − κ + 1  κN  ,  γ2 = 1 + lim  N→∞  b − κ + 1  κN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='88)  This can be understood from [49, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)] in the Cauchy case, and the loop equation  analysis done in [319], [142], otherwise (see also references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The classical β ensembles have been studied extensively throughout the last couple of  decades, leading to a vast collection of results: see [222], [80], [280], [43], [319], [218] and  references therein for a diverse but non-exhaustive sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A result that is of great use to  us is that the spectral moments of the classical β ensembles can be expanded in either N or  1/N, in line with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 (Dumitriu–Edelman ’06, Dumitriu–Paquette ’12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let mk be the kth spectral moment  of the Gaussian, Laguerre, or Jacobi β ensemble, as specified by equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the  first two cases, mk is a degree-(k + 1) polynomial in N (identically zero in the Gaussian case with k  odd due to the weight w(G)(λ) being an even function) [94], while in the Jacobi case, mk/N has a  large N expansion in 1/N [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For later use, we implicitly define the (N-independent) polynomial  and series coefficients Mk,l as follows:  m(G)  2k =  k  ∑  l=0  M(G)  k,l N1+k−l,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='89)  m(L)  k  =  k  ∑  l=0  M(L)  k,l N1+k−l,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='90)  m(J)  k  =  ∞  ∑  l=0  M(J)  k,l N1−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='91)  The existence of these expansions was established in [94], [95] using Jack polynomial  theory (see also the later work [242] and the discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Cauchy and  shifted Jacobi spectral moments m(Cy)  k  , m(sJ)  k  can be written in terms of the Jacobi spectral  moments m(J)  k  (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 below) so that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='91)  translates to a similar expansion for the m(Cy)  k  , m(sJ)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Classical even-β ensembles  The methods used in Chapter 2 are not suitable for treating the classical general-β ensembles,  but do extend past the β = 1, 2, 4 paradigm corresponding to the orthogonal, unitary, and  39  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  symplectic ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, we are technically able to study a scheme in between these two  which we will call the classical even-β ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These are simply the classical β ensembles  with β constrained to be a positive even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The significance of β being even is that the  Vandermonde factor |∆N(λ)|β in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) is then a polynomial in the eigenvalues  {λi}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This was seen to be a critical requirement for some results related to Selberg  correlation integrals, the 1/r2 quantum many-body system, and the Calogero–Sutherland  model (see [119, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2], [123, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, the works [116], [81],  [120], [133] on various subsets of the classical even-β ensembles contain (edge scalings of)  expressions for the relevant eigenvalue densities ρ(λ) in terms of hypergeometric functions  and β-dimensional integrals, which we use as points of comparison in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  One issue with studying the classical even-β ensembles is that the case β = 1 is not  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As outlined earlier, this case is very important due to it corresponding to real  symmetric matrices with orthogonal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that we are able to extend our  results in Chapter 2 to the classical β ensembles with β such that 4/β is an even integer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' this  allows for treatment of the β = 1 case, along with classical β ensembles corresponding to  some non-integer rational values of β such as β = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This extension is possible because the  works [94], [95] cited in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 tell us that the coefficients Mk,l therein are palindromic  polynomials in −1/κ = −2/β, upon appropriate scaling of the parameters a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be  precise, since the even-β analysis in Chapter 2 focuses on the spectral moments mk and their  generating functions W1(x), it is amenable to the following β ↔ 4/β duality relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us highlight the dependence of the spectral moments mk (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) and resolvents  W1(x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) on the parameters N, κ = β/2, a, b, α by listing them as arguments (recall that we  write the parameter η in the Cauchy weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) as η = −κ(N − 1) − 1 − α with α ∈ C and  Re(α) > −1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the kth spectral moments of the classical β ensembles satisfy the duality  relations [94], [95] (see also [149, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A] by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Borodin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Gorin)  m(G)  2k (N, κ) = (−κ)k−1m(G)  2k (−κN, 1/κ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='92)  m(L)  k (N, κ, a) = (−κ)k−1m(L)  k (−κN, 1/κ, −a/κ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='93)  m(J)  k (N, κ, a, b) = −κ−1m(J)  k (−κN, 1/κ, −a/κ, −b/κ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='94)  m(Cy)  k  (N, κ, α) = −κ−1m(Cy)  k  (−κN, 1/κ, −α/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='95)  40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  These extend to the duality relations [319], [142]  W(G)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, κ) = −iκ−3/2W(G)  1  (ix/√  κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' −κN, 1/κ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='96)  W(L)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, κ, a) = κ−2W(L)  1  (−x/κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' −κN, 1/κ, −a/κ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='97)  W(J)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, κ, a, b) = −κ−1W(J)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' −κN, 1/κ, −a/κ, −b/κ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='98)  W(Cy)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, κ, α) = −κ−1W(Cy)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' −κN, 1/κ, −α/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='99)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The references cited above do not study the Cauchy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, the duality  relations for the Jacobi β ensemble translate to the Cauchy β ensemble through the identities  presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2: Identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) expresses the spectral moments m(sJ)  k  of the shifted  Jacobi β ensemble (the classical β ensemble with weight w(sJ)(λ) = (1 − λ)a(1 + λ)bχ−1<λ<1,  as introduced at the end of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) in terms of the m(J)  k  above, which shows that m(sJ)  k  satisfies  the duality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 combined with Carlson’s theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the proof of  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) shows that  m(Cy)  k  (N, κ, α) = (−1)k/2m(sJ)  k  (N, κ, a, b)  ���  a=b=−κ(N−1)−1−α,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='100)  so that equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='94) implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain the duality relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='99), one can  repeat these arguments with either equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20) specifying the resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Matrix Product Ensembles  From the viewpoint of using random matrices to model transformations such as quantum  operators, scattering channels, or evolution operators for certain systems, it is quite natural  to use products of random matrices to model repeated applications of such transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Hence, the theory of matrix product ensembles has applications in the study of wireless  communications [253], [301], quantum transport [33], the stability and chaoticity of large  dynamical systems [69], [184], finance [47], and the stability of neural networks [272], [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  There are also interesting connections to Fuss–Catalan and Raney numbers [273], [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Investigations into products of random matrices can be traced back to the 1950s, with  many fundamental results and techniques being developed in the works [153], [175], [147],  [270], [278], [263] (among others) carried out over the following four decades;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see [177]  (group theory) and [69] (statistical physics) for textbook treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There has since been  41  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  a consistent level of research on products of random matrices (some notable works being  [189], [191] and the applications cited above, along with the inception and success of free  probability theory [306], [244]), but the last decade has seen a remarkable surge in interest on  this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The beginning of this latter era is marked by the derivation of correlation kernels  (recall Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) encapsulating detailed knowledge of the eigenvalue and singular value  statistics of certain matrix product ensembles, where the products contain a finite number of  factors and each factor is a finite-sized matrix — most results from before this era pertain to  regimes where either the matrix sizes or number of factors are taken to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The reader  is referred to [8], [182] for reviews on this period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  At the beginning of this chapter, it was said that two types of random matrix ensembles  will be studied in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The first is the family of classical matrix ensembles reviewed in  the previous section, while the second consists of the particular matrix product ensembles  that we now introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let m, N0 ∈ N, fix ν0 := 0, ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm ∈ N, and define Ni := N0 + νi with  N := Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 of the Ginibre and Gaussian ensembles, let H  be an N0 × N0 GUE matrix and for 1 ⩽ i ⩽ m, let Gi be drawn independently from the  Ni−1 × Ni complex Ginibre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the N × N product  Hm := G†  m · · · G†  1HG1 · · · Gm  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  represents the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) Hermitised matrix product ensemble [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When m = 1, we  alternatively say that H1 represents the (N0, N) Hermitised Laguerre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let m, N0/2 ∈ N, fix ν0 := 0, ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm ∈ N, and define Ni := N0 + 2νi  with N := Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 of the Ginibre ensembles, let JN0 be the elementary  antisymmetric matrix defined by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) and for 1 ⩽ i ⩽ m, let Gi be drawn  independently from the Ni−1 × Ni real Ginibre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the N × N product  Jm := GT  m · · · GT  1 JN0G1 · · · Gm  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  represents the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) antisymmetrised matrix product ensemble [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When m = 1,  we alternatively say that J1 represents the (N0, N) antisymmetrised Laguerre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We think of these ensembles as perturbations of the more widely studied Wishart product  ensembles [8], [182], which are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10, let m, N0 ∈ N, fix ν0 := 0, ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm ∈ N, and define  Ni := N0 + νi with N := Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For 1 ⩽ i ⩽ m, let Gi be drawn independently from the  Ni−1 × Ni real, complex, or quaternionic Ginibre ensemble, as specified by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  (fixing the number system across all Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the N × N product  Wm := G†  m · · · G†  1G1 · · · Gm  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  represents the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) real, complex, or quaternionic Wishart product ensemble, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Setting m = 1 recovers the (N0, N) Laguerre ensemble, in keeping with Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Our present goal in studying the Hermitised and antisymmetrised matrix product ensem-  bles is to better understand the ramifications of perturbing the Wishart product ensembles  in the corresponding ways (inserting a GUE matrix H or elementary antisymmetric matrix  JN0 in the middle of the relevant products), in terms of both changes to the eigenvalue  statistics and changes to the effectiveness of existing techniques for analysing these statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In particular, we would like to see how the recent loop equation analysis of Dartois and  Forrester [74] for the (N, N, N) complex Wishart product ensemble extends to the Hermi-  tised and antisymmetrised matrix product ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in Chapter 4, we derive loop  equations characterising the correlator expansion coefficients Wl  n of the Hermitised and  antisymmetrised Laguerre ensembles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the product ensembles of Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11  with m = 1) after they have been scaled such that their connected n-point correlators Wn  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) admit large N expansions of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) — we argue the validity of these large  N expansions in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 by giving combinatorial interpretations (which are interesting in  their own right) to the mixed cumulants generated by the Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our decision to restrict the  analysis of Chapter 4 to the m = 1 ensembles stems from the fact that taking m to be any  larger results in loop equations that are too unwieldy to fit within the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Nonetheless, the loop equations derived in Chapter 4 have rich enough structure for us to  make meaningful comparison with the loop equations presented in [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We are also able to  glean insight on the expected structure of the loop equations pertaining to arbitrary m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Exploring the consequences of perturbing the Wishart product ensembles is not our only  motivation for studying the Hermitised and antisymmetrised matrix product ensembles,  as this could be done by considering any number of alternative, comparable perturbations  of said ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rather, our secondary motivation for studying the Hermitised and  43  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  antisymmetrised matrix product ensembles is that they have recently been shown to be  related to certain Muttalib–Borodin ensembles and a generalisation, respectively analogue,  of the Harish-Chandra–Itzykson–Zuber (HCIZ) integral [143], [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, we are able  to compare the results of Chapter 4 against explicit functional forms for the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the Hermitised and antisymmetrised matrix product ensembles that were obtained  in [143], [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, it is expected that the results of Chapter 4 will help advance  our understanding of the broader connection between products of random matrices and  Muttalib–Borodin ensembles that is seen in the literature [216], [141], [182], [321], [135], [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us now review, in order, the complex Wishart product ensembles, Muttalib–Borodin  ensembles, and HCIZ-type integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We keep our review brief by focusing mostly on aspects  of these topics that connect them to the ensembles that are studied in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Complex Wishart product ensembles and their correlation kernels  If one wishes to capture the essence of multiplying generic random matrices, it is reasonable  to study products of Ginibre matrices: On one hand, prescribing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s on the matrix factors  makes the situation relatively tractable and allows the use of some powerful analytical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  On the other hand, there is no benefit in imposing symmetry constraints on the involved  matrices, since products of self-adjoint matrices are not necessarily self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, a  large portion of recent work in the field has been on products of Ginibre matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see  [58], [7], [217], [183] and references therein for a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that products of truncated  Haar-distributed unitary matrices (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the first point of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and products involving  inverse Ginibre matrices have also drawn considerable interest [185], [13], [122], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In studying the spectral properties of products of Ginibre matrices, one is interested  in either the eigenvalues, which are generally complex, or the singular values, which are  non-negative real (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Both have been studied in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [7], [185], [3] and [12],  [11], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Due to their relative simplicity, our interest lies in the squared singular  values, which are equivalent to the eigenvalues of the Wishart product matrices Wm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like the eigenvalues of Wm, the eigenvalues of the matrix product Hm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  are real, since it is Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In a similar vein, the eigenvalues of the antisymmetric matrix  Jm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) lie on the imaginary axis in complex conjugate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  Let us now present the analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 for the complex Wishart product  ensembles, which we obtain by combining results of [11] and [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) be the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the non-zero eigenvalues of the complex  Wishart product matrix Wm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and let us recall [31] that the Meijer G-function is defined as  Gm,n  p,q  �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ap  b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , bq  ���� z  �  :=  1  2πi  �  γ  ∏m  j=1 Γ(bj + u) ∏n  j=1 Γ(1 − aj − u)  ∏  q  j=m+1 Γ(1 − bj − u) ∏  p  j=n+1 Γ(aj + u)z−u du,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  where the choice of integration contour γ is too involved to specify here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for 1 ⩽ k ⩽ N0, the  k-point correlation function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='72)) associated with p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) has the form  ρ(cWm)  k  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N0) :=  N0!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (N0 − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  �  RN0−k p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) dλk+1 · · · dλN0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  = Det  �  K(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(λi, λj)  �k  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  with correlation kernel given by  K(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(x, y) =  � 1  0 G1,0  1,m+1  �  N0  0, −ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , −νm  ���� ux  �  Gm,1  1,m+1  �  −N0  ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm, 0  ���� uy  �  du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) tells us that the complex Wishart product ensembles are determinantal  point processes, much like what was seen in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 for the β = 2 classical matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  However, while this property of the β = 2 classical matrix ensembles is a consequence of  their being orthogonal polynomial ensembles, the same is not true for the complex Wishart  product ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, the complex Wishart product ensembles, together with the  Hermitised and antisymmetrised matrix product ensembles, are biorthogonal ensembles [38]  (see the next subsection for a formal definition of this term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, all three of these matrix  product ensembles belong to a further subclass of biorthogonal ensembles that are simply  called polynomial ensembles4 [216];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' these so-called polynomial ensembles are characterised by  their eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s having the form  p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) =  1  NN0  ∆N0(λ) Det  �  wj−1(λi)  �N0  i,j=1 ,  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0 ∈ R,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  4In the case of the antisymmetrised matrix product ensembles, it is common practice to study the positive  eigenvalues of iJm, since these are real and independent, in keeping with Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Technically, it is the  positive eigenvalues of iJm that constitute a biorthogonal ensemble and it is the squares of these eigenvalues  that exhibit the structure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) of polynomial ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  45  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  where NN0 is a normalisation constant, ∆N0(λ) is the Vandermonde determinant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), and  the family of weight functions {wj−1(λ)}N0  j=1 is such that the above j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) and p(Hm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) governing the non-zero  eigenvalues of the complex Wishart product matrix Wm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and Hermitised matrix product Hm  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), respectively, are given by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) with weights [11], [143]  w(cWm)  j  (λ) = Gm,0  0,m  �  −  ν1 + j, ν2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm  ���� λ  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  w(Hm)  j  (λ) = (sgn λ)j  m  ∏  i=1  2νi−1  √π G2m+1,0  0,2m+1  �  −  ν1  2 , ν1+1  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm  2 , νm+1  2  , j  2  ����  λ2  4m  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  where we recall the definition of the Meijer G-function from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similarly, the positive  eigenvalues of iJm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) are distributed according to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [144]  p(iJm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0/2) =  1  N (iJm)  N0/2  ∏  1⩽k<l⩽N0/2  (λ2  l − λ2  k) Det  �  w(iJm)  j−1 (λi)  �N0/2  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  w(iJm)  j  (λ) = Gm,0  0,m  �  −  2ν1 + j, 2ν2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 2νm  ���� λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  The associated normalisation constants can be found in [11], [143], [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The eigenvalue statistics of the matrix products considered in the above proposition are  unchanged upon replacing the rectangular real and complex Ginibre matrices Gi (1 ⩽ i ⩽ m)  therein by corresponding induced Ginibre matrices ˆGi [112], [11], [185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These matrices ˆGi are  drawn from the space of N0 × N0 real, respectively complex, matrices with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(indG)( ˆGi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N0, β, νi) =  1  Z(indG)  N0,β,νi  Det( ˆG†  i ˆGi)βνi/2 exp  �  − Tr( ˆG†  i ˆGi)  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  where Z(indG)  N0,β,νi is a normalisation constant and β = 1, 2 refers to the real and complex  cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The fact that making the replacements described above does not affect  the eigenvalue statistics of interest was used in the proof of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) given in [11],  where reformulating the product (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) in terms of square induced Ginibre matrices en-  abled the use of the HCIZ integral formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If one interprets p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0),  p(Hm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0), and p(iJm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0/2) as eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the products Wm (in  46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  the complex case), Hm, and iJm rewritten in terms of the induced Ginibre matrices described  above, then Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 hold true for general real parameters ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the discussion surrounding Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, the results of these propositions  are invariant under permutations of the parameters ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm due to the weak commutation  relation established in [185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A point of intrigue regarding the antisymmetrised matrix product ensembles is that they  have recently been shown in [144] to be closely related to the complex Wishart product  ensembles generalised through the induced Ginibre matrices described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let Jm be as specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 and let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0/2 denote the positive  eigenvalues of iJm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, by [144, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2], the variables λ′  j = λ2  j /4m (1 ⩽ j ⩽ N0/2) are  statistically equivalent to the eigenvalues of the product ˆG†  2m · · · ˆG†  1 ˆG1 · · · ˆG2m of (N0/2) × (N0/2)  induced Ginibre matrices with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  P(indG)( ˆG2i−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N0/2, 2, νi),  P(indG)( ˆG2i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N0/2, 2, νi − 1/2),  1 ⩽ i ⩽ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In short, the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the positive eigenvalues of iJm is given by  p(iJm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0/2) = 2N0(1/2−m)λ1 · · · λN0/2  × p(cW2m)(λ2  1/4m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λ2  N0/2/4m)  ���� ν2i−1�→νi,  ν2i�→νi−1/2  �m  i=1  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  where the right-hand side can be read off by reparametrising equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In addition to highlighting a relationship between the weights w(cWm)  j  (λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and  w(iJm)  j  (λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12), the above proposition tells us that the positive eigenvalues of iJm constitute  a determinantal point process whose k-point correlation functions have the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) with  correlation kernel given by  K(iJm)  N0/2,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(x, y) = 21−2m√xy K(cW2m)  N0/2,ν1,ν1−1/2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm,νm−1/2(x2/4m, y2/4m),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  where the right-hand side is specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us mention here that it was  shown in [143] that the non-zero eigenvalues of the Hermitised matrix product Hm also  constitute a determinantal point process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we do not present the relevant correlation kernel  here, but note that it is made explicit in [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  47  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Scalings of the eigenvalue densities and correlation kernels  Recall from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 that in the case of the classical matrix ensembles, the N → ∞  limits ρ0(λ) of the global scaled eigenvalue densities ˜ρ(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) have distinct forms for  each classical weight with, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the Gaussian and Laguerre weights corresponding to the  Wigner semi-circle and Marˇcenko–Pastur laws, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that the analogous  limiting densities (now taking N0 → ∞) of the matrix product ensembles considered in this  section relate to the Fuss–Catalan distribution [273], [182] (see also the independent works  of Biane [35, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A] and Neuschel [262] for a Plancherel–Rotach-like parametrisation in  terms of elementary functions)  ρ(FCm)(λ) =  1  √  2π  mm−3/2  (m + 1)m+1/2 Gm,0  m,m  �  1  m, 0, − 1  m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , − m−2  m  −  1  m+1, −  2  m+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , −  m  m+1  ����  mm  (m + 1)m+1 λ  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  which is so named due to its spectral moments being the Fuss–Catalan numbers [167]:  m(FCm)  k  :=  �  R λkρ(FCm)(λ) dλ =  1  mk + 1  �(m + 1)k  k  �  ,  k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  Note that due to properties of the Meijer G-function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), the density ρ(FCm)(λ) has compact  support [0, (m + 1)m+1/mm], on which it integrates to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let ρ(cWm)(λ), ρ(Hm)(λ), and ρ(iJm)(λ) denote the eigenvalue densities  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10), equivalently the 1-point correlation functions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), associated with the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0), p(Hm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0), and p(iJm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0/2) specified in Propo-  sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, in analogy with Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6, and partially following [182],  [143], define the corresponding global scaled eigenvalue densities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) to be  ˜ρ(cWm)(λ) := Nm−1  0  ρ(cWm)(Nm  0 λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  ˜ρ(Hm)(λ) :=  1  √  2 Nm−1/2  0  ρ(Hm)( 1  √  2 Nm+1/2  0  λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  ˜ρ(iJm)(λ) := 2Nm−1  0  ρ(iJm)(Nm  0 λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  Furthermore, recall the notation ρ0(λ) = limN0→∞ ˜ρ(λ) introduced in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It has been shown in [29], [273], [143] that for fixed ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1, the large N0 limits  of ˜ρ(cWm)(λ) and ˜ρ(Hm)(λ) are given in terms of the Fuss–Catalan distribution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) by  ρ(cWm),0(λ) = ρ(FCm)(λ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  ρ(Hm),0(λ) = |λ|ρ(FC2m+1)(λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  48  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  Setting x = y in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) so that it may be read as a relation between the correspond-  ing eigenvalue densities, and then combining this relation with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) shows that  (see also [127] for the m = 1 case)  ρ(iJm),0(λ) = 2λρ(FC2m)(λ2)χλ>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  It can immediately be seen that, aside from subtleties concerning the parameter m, ρ(Hm),0(λ)  is essentially the symmetric extension of ρ(iJm),0(λ) to the negative real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover,  both of these densities relate to ρ(cWm),0(λ) via the mapping λ �→  √  λ, much like how the  Wigner semi-circle (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) and Marˇcenko–Pastur (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) laws are related by the equality  ρ(G),0(λ) = 2|λ| ρ(L),0(2λ2) when the Laguerre parameter a is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, recall that  ρ(cW1),0(λ) = ρ(L),0(λ) and ρ(H0),0(λ) = ρ(G),0(λ/  √  2), by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The global scaling regimes just discussed are such that the limiting densities ρ0(λ) have  compact connected supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In contrast, one is also interested in local scaling regimes, where  the eigenvalue density ρ(λ) is centred on a particular point λ0 ∈ supp ρ and then scaled so  that the spacing between this eigenvalue and its neighbours is of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The resulting  statistics depends on whether λ0 is positioned at a soft edge, a hard edge, or within the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the case of the complex Wishart product ensembles with ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1 fixed, λ0 is said  to be at the soft edge if limN0→∞ λ0/Nm  0 = (m + 1)m+1/mm, the hard edge if limN0→∞ λ0/Nm  0  vanishes, and in the bulk if limN0→∞ λ0/Nm  0 ∈ (0, (m + 1)m+1/mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (The analogous scaling  regimes of ρ(Hm)(λ) and ρ(iJm)(λ) are likewise defined by comparing similar scaled limits,  with scalings prescribed by equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20), to ±(m + 1)(m+1)/2/mm/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let m ∈ N and ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, let  x0 ∈ (0, (m + 1)m+1/mm),  cb = Nm−1  0  /ρ(FCm)(x0),  x+ =  �  N0  m  �m  (m + 1)m+1,  c+ = Nm−2/3  0  21/3mm−1  (m+1)m+2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  Then, the hard edge, bulk, and soft edge limiting forms of the correlation kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) are respectively  given by [228], [217]  lim  N0→∞  1  N0  K(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm (x/N0, y/N0) = K(Meijer)  ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm (x, y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  lim  N0→∞ cbK(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm (cbx + Nm  0 x0, cby + Nm  0 x0) = K(sine)(x, y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  lim  N0→∞ c+K(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm (c+x + x+, c+y + x+) = K(Airy)(x, y),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  49  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  where  K(Meijer)  ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm (x, y) =  � 1  0 G1,0  0,m+1  �  −  0, −ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , −νm  ���� ux  �  Gm,0  0,m+1  �  −  ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm, 0  ���� uy  �  du,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  K(sine)(x, y) = sin π(x ��� y)  π(x − y)  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  K(Airy)(x, y) = Ai(x)Ai′(y) − Ai′(x)Ai(y)  x − y  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  are the Meijer G-, sine, and Airy kernels, respectively (recall that the Meijer G-function is partially  defined in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and that Ai(x) denotes the Airy function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall from the discussion following Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 that the hard edge of  the Laguerre ensembles degenerates to a soft edge upon letting the Laguerre parameter a  grow linearly with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A similar phenomenon is exhibited by the complex Wishart product  ensembles [11], whose hard edge ceases to be if all of the parameters ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm scale with  N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If one writes νi = ˆνiN0 + δi with ˆνi, δi = O(1) for 1 ⩽ i ⩽ m and further enforces  ˆν1 = · · · = ˆνl = 0, but ˆνl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ˆνm > 0 for some 1 ⩽ l ⩽ m, then it is known from [182] that  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) should be replaced with  lim  N0→∞ c−K(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(c−x, c−y) = K(Meijer)  ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νl (x, y),  c− = ˆνl+1 · · · ˆνmNm−l−1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  It follows from Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9 that the local correlations of the positive eigen-  values of iJm are characterised by the Meijer G-, sine, and Airy kernels, in the appropriate  scaling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, it is mentioned (without proof) in [143] that the bulk and soft edge  scaling limits of the Hermitised matrix product ensembles are also described by the sine and  Airy kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More interesting, however, is the microscopic behaviour of these ensembles at  the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There, one observes the so-called double-sided hard edge phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let m ∈ N and ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, let K(Hm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(x, y) denote  the correlation kernel corresponding to p(Hm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0) in analogy with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  for x, y ∈ R \\ {0}, one has the limit [143]  lim  N0→∞  1  √  N0/2K(Hm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm  �  x  √  N0/2,  y  √  N0/2  �  = sgn(xy)|x|  4m  K(Meijer)  1  2, ν1  2 , ν1+1  2  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', νm  2 , νm+1  2  � x2  4m , y2  4m  �  + |y|  4m K(Meijer)  − 1  2 , ν1  2 , ν1−1  2  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', νm  2 , νm−1  2  � x2  4m , y2  4m  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is important to note that the term ‘double-sided hard edge’ mentioned  above merely refers to the fact that the local correlations at the origin of the spectra of  the Hermitised matrix product ensembles relate to the hard edge of the complex Wishart  product ensembles via the mapping λ �→ λ2 and that said spectra do not actually exhibit  an edge at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, it is shown in [143] that setting m = 0 in the right-hand side  of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) recovers the sine kernel, which corresponds to bulk statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us  also take this opportunity to point out that the limiting densities ρ(cWm),0(λ) of the complex  Wishart product ensembles do not always have inverse square root singularities at the hard  edge, in contrast to what is known for the classical matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rather, in the setting  of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11, ρ(cWm),0(λ) diverges like λ−l/(l+1) as λ → 0 [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It is well known [115], [39], [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 7] that the local bulk, soft edge, and hard edge  statistics of the β = 2 classical matrix ensembles are respectively governed by the sine (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29),  Airy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30), and Bessel kernels (the required edge scalings are given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), with the  latter defined as  K(Bessel)  a  (x, y) = Ja(√x)√yJ′  a(√y) − √xJ′  a(√x)Ja(√y)  2(x − y)  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  where Ja(x) is the Bessel function and a is the exponent seen in the Laguerre and Jacobi  weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the microscopic bulk and soft edge statistics of the matrix product en-  sembles discussed in this section are within the same universality classes as those pertaining  to the β = 2 classical matrix ensembles, but this is not the case at the hard edge (although,  the Meijer G-kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) reduces to the Bessel kernel upon setting m = 1, as one would  expect [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this way, as well as their being biorthogonal ensembles rather than orthogonal  polynomial ensembles, our matrix product ensembles display universal structures that are  fundamentally different to those seen in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Muttalib–Borodin and biorthogonal ensembles  In their 1988 work [239] on multichannel disordered wires, Mello, Pereyra, and Kumar  (building on the 1982 work [87] of Dorokhov) derived a Fokker–Planck equation that came  to be known as the DMPK equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This equation characterises the eigenvalues of the  associated transfer matrix, which encode physical statistics of the wire at hand, such as its  51  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  quantum conductance and electron localisation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The DMPK equation was shown in  the 1993 work [34] of Beenakker and Rejaei to be solved by the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(DMPK)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) =  1  N (DMPK)  N,s  N  ∏  i=1  exp (−VDMPK(λi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' s)) ∆N(λ)  ×  ∏  1⩽j<k⩽N  �  arcsinh2 λ1/2  k  − arcsinh2 λ1/2  j  �  ,  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  where ∆N(λ) is the Vandermonde determinant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), s is a length scale, N (DMPK)  N,s  is a  normalisation constant, and  VDMPK(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' s) = N  s arcsinh2(  √  λ)  �  1 + O(N−1)  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  is a background potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Two years later, Muttalib proposed [254] the study of simplifica-  tions of the above model that correspond to eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the form  p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) =  1  NN,θ  N  ∏  i=1  exp (−V(λi)) ∆N(λ)  ∏  1⩽j<k⩽N  (λθ  k − λθ  j ),  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  where NN,θ is a normalisation constant, θ ∈ N is a deformation parameter (the ensembles  can be thought of as deformations of the θ = 1 cases, which correspond to the OPEs  discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), and V(λ) is a general background potential such that the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  is well-defined — this structure relates to the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) in the limit θ → 0+ [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It turns out that the relation between Muttalib’s model and the DMPK equation was not  pursued much further by the physics community, but varying forms of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  appeared nonetheless in a range of works [230], [66], [65], [216], [127] over the following two  decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A significant motivation for many of these works was that the associated models  were shown to be exactly solvable by Muttalib [254] and Borodin [38]: Muttalib used the  Konhauser theory of biorthogonal polynomials [209] (this theory was later realised [15] to  have been studied much earlier in [84], [79]) to prove that the ensembles corresponding  to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) are determinantal point processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Borodin extended this work by  introducing the so-called biorthogonal Hermite, Laguerre, and Jacobi ensembles (see below)  and giving explicit formulae for their correlation kernels (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in  the recent work [135], the name Muttalib–Borodin ensemble was given to the systems of  positive real eigenvalues governed by j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36), with θ now a positive real  deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Having fixed N ∈ N, a biorthogonal ensemble [38] is a system of eigenvalues  {λi}N  i=1 supported on a possibly infinite interval I ⊆ R with eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the form  p(b,w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) =  1  N (b,w)  N  N  ∏  i=1  w(λi) Det  �  fj(λi)  �N  i,j=1 Det  �  gj(λi)  �N  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  where N (b,w)  N  is a normalisation constant, w(λ) is an admissible weight supported on I (that  is, w(λ) is continuous, non-negative, and real-valued over I [209];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8), and  the sequences of real-valued functions { fj(λ)}N  j=1 and {gj(λ)}N  j=1, along with w(λ), are such  that the above j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Polynomial ensembles are examples of biorthogonal ensembles since setting  N = N0,  w(λi) = 1,  fj(λi) = λj−1  i  ,  gj(λi) = wj−1(λi)  in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) recovers the structure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, Muttalib–Borodin ensembles  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) can be seen to be biorthogonal ensembles by setting  w(λi) = exp(−V(λi)),  fj(λi) = λj−1  i  ,  gj(λi) = λθ(j−1)  i  in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Rewriting equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) by absorbing the factors exp(−V(λi))  into the determinant ∏1⩽j<k⩽N(λθ  k − λθ  j ) shows that Muttalib–Borodin ensembles are also  polynomial ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The adjective ‘biorthogonal’ in the above definition refers to the fact that, after reordering  the sequences { fj(λ)}N  j=1 and {gj(λ)}N  j=1 if necessary (consider the canonical generalisation  of [38, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5] involving LUP decompositions [195, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3]), there exist further sequences  of so-called biorthogonal functions [38] (biorthogonal polynomials in the case of Muttalib–Borodin  ensembles, among others [209]) {ξj(λ)}N  j=1 and {ηj(λ)}N  j=1, along with a sequence of finite,  positive constants {rj}N  j=1, such that for each 1 ⩽ i, j ⩽ N,  ξj(λ) − fj(λ) ∈ Span  �  f1(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , fj−1(λ)  �  ,  ηj(λ) − gj(λ) ∈ Span  �  g1(λ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , gj−1(λ)  �  ,  and (recalling that the indicator function χA equals one when A is true and zero otherwise)  �  I ξi(λ)ηj(λ)w(λ) dλ = rjχi=j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  53  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Using the invariance of determinants under row and column operations in a similar  fashion to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) can be rewritten as [254], [38]  p(b,w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) =  1  N (b,w)  N  N  ∏  i=1  w(λi) Det  �  ξj(λi)  �N  i,j=1 Det  �  ηj(λi)  �N  i,j=1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  = Det  �  K(b,w)  N  (λi, λj)  �N  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  where {ξj(λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ηj(λ)}N  j=1 is the system of biorthogonal functions described above and  K(b,w)  N  (x, y) :=  �  w(x)w(y)  N−1  ∑  k=0  1  rk  ξk(x)ηk(y)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  is the correlation kernel obtained by interchanging the indices i ↔ j in the second determi-  nant of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) and then combining all factors therein into a single determinant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It was shown by Muttalib [254] that the biorthogonality relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  implies that the above correlation kernel has the reproducing property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in  analogy with equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='73) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), the k-point correlation function (1 ⩽ k ⩽ N)  corresponding to p(b,w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) is given by  ρ(b,w)  k  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) :=  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (N − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  �  RN−k p(b,w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) dλk+1 · · · dλN  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  = Det  �  K(b,w)  N  (λi, λj)  �k  i,j=1 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43)  so that biorthogonal ensembles fall within the class of determinantal point processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As mentioned earlier, the biorthogonal ensembles at the focus of Borodin’s work [38]  were the biorthogonal Hermite, Laguerre, and Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In present day terminology  [125], these correspond to the Hermite, Laguerre, and Jacobi Muttalib–Borodin ensembles  (the latter with parameter b set to zero), which have eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the form  p(w,θ)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) =  1  N (w)  N,θ  N  ∏  i=1  w(λi) ∆N(λ)  ∏  1⩽j<k⩽N  ((sgn λk)|λk|θ − (sgn λj)|λj|θ)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  defined on RN, with respective (semi-)classical weights (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9))  w(λ) =  �  �  �  �  �  �  �  �  �  �  �  �  �  |λ|ae−λ2,  generalised Hermite,  λae−λχλ>0,  Laguerre,  λa(1 − λ)bχ0<λ<1,  Jacobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45)  54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  Here, N (w)  N,θ is again a normalisation constant and θ > 0, a, b > −1 are real parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As one might expect, the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) is a generalisation of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' they are  equivalent if all eigenvalues are positive and/or if θ is an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The correlation kernels (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) governing the statistics of the Muttalib–Borodin ensembles  with weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) (with b = 0) were made explicit in [38], along with their (double-sided)  hard edge limiting forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The associated biorthogonal polynomials were studied earlier in  [210], [233], [296].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More recently, Forrester and Ipsen [125] have shown how Selberg integral  theory can be used to obtain explicit formulae for the biorthogonal polynomials related to  the weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), in addition to the so-called Jacobi prime, generalised symmetric Jacobi,  and generalised Cauchy weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Incidentally, the biorthogonal functions relating to the  correlation kernels K(cWm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(x, y), K(iJm)  N0/2,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(x, y), and K(Hm)  N0,ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm(x, y) discussed in the  previous subsection have also been made explicit in the works [11], [143], [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Relations to matrix product ensembles  Our interest lies in the Laguerre and Hermite Muttalib–Borodin ensembles, as they have  been shown [141], [182], [143] to approximate the complex Wishart and Hermitised matrix  product ensembles upon applying the asymptotic formula [231, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7]  Gm,0  0,m  �  −  ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm  ���� λ  �  ∼  λ→∞  1  √m  � 2π  λ1/m  �(m−1)/2  λ(ν1+···+νm)/pe−mλ1/m �  1 + O(λ1/m)  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  to the Meijer G-functions in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and appropriately changing variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let m, N0 ∈ N and ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1 be fixed (with N0 even when working  with iJm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, let p(H,θ)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a) and p(L,θ)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a) denote the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) with w(λ) being the generalised Hermite and Laguerre weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Then, substituting the large λ approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) into the formulae of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 shows that  in the λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0 → ∞ limit with ¯ν := ν1 + · · · + νm and γi = λi/√  2m + 1 (1 ⩽ i ⩽ N0),  p(cWm) ��  λ1  m  �m  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ,  � λN0  m  �m�  ≈  N0  ∏  i=1  �  λi  m  �1−m  p(L,m) �  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ¯ν + m−1  2  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  p(iJm) ��  λ1  m  �m  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ,  � λN0/2  m  �m�  ≈  N0/2  ∏  i=1  �  λi  m  �1−m  p(L,2m) �  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2¯ν + m−1  2  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48)  p(Hm) �  2mγ2m+1  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 2mγ2m+1  N0  �  ≈  N0  ∏  i=1  γ−2m  i  2m√  2m + 1  p(H,2m+1) (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2¯ν + m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49)  55  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is shown in [144, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4] that the approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48) is in fact exact  for m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This complements the fact that the approximations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) are also  exact for m = 1 and m = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (Comparing equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) to equations  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) reveals that p(L,1)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a) and p(H,1)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 0) are respectively the  eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the Laguerre and Gaussian unitary ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')  The works [141], [182] contain formulae similar to the approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47), while the  approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) is given in [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, the approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) represents the  success of the latter work in relating the Hermite Muttalib–Borodin ensembles to products  of random matrices with explicit p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s on their entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, [143] completed the task of  finding matrix model realisations of the Muttalib–Borodin ensembles with (semi-)classical  weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), supplementing analogous realisations of the Laguerre and Jacobi Muttalib–  Borodin ensembles given in the earlier works [5], [141], [135], [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We note, in particular, that  the Jacobi Muttalib–Borodin ensembles relate to products of Ginibre matrices and truncations  of Haar-distributed unitary matrices (see [205], [135, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In light of the scalings discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, a question of obvious interest regarding  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 is, “How accurate are the approximations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) in the large N0  limit and where do they break down?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In other words, “When taking N0 → ∞, how do the  eigenvalue statistics of the matrix product ensembles discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 differ from those  of the Muttalib–Borodin ensembles that they relate to via Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out  that the global, bulk, and soft edge scaling statistics of the Muttalib–Borodin ensembles  pertaining to the right-hand sides of the approximations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) are equivalent to  those of the corresponding matrix product ensembles [38], [327], [127], [135], [143] — that is,  the respective statistics are yet again described by the Fuss–Catalan distribution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), the  sine kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29), and the Airy kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This makes sense at a heuristic level since,  by the discussion preceding Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9, the eigenvalues of the matrix product ensembles  considered in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 that are not at the (double-sided) hard edge grow like Nm  0  and are thus large enough, in the N0 → ∞ limit, to fall within the scope of said proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As one might expect, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 breaks down at the origin, but not too drastically:  Let K(L,θ)  N  (x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a) denote the correlation kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) corresponding via equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) to  the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(L,θ)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a) specified in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Borodin [38] showed  56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  that the hard edge scaling limit of this kernel is given by  K(a,θ)(x, y) := lim  N→∞ N−1/θK(L,θ)  N  (N−1/θx, N−1/θy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a)  = θxa  � 1  0 J(a+1)/θ,1/θ(xt) Ja+1,θ((yt)θ)ta dt,  where  Ja,b(x) =  ∞  ∑  j=0  (−x)j  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='Γ(a + jb)  is Wright’s generalised Bessel function [324].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It has been shown [216, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1] for θ ∈ N,  as relevant to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 (they also show an analogous result for 1/θ ∈ N that we do  not display here), that  x1/θ−1K(a,θ)(θx1/θ, θy1/θ) = K(Meijer)  a+1  θ  −1, a+2  θ  −1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', a+θ  θ  −1(y, x),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50)  where K(Meijer)  ν1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',νm (x, y) is the Meijer G-kernel (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) describing the hard edge statistics of the  complex Wishart product ensembles via equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Hermite case, Borodin  [38] showed that scaling the correlation kernel corresponding to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(H,θ)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a) to the origin results in a limiting kernel that is related to K(a,θ)(x, y) in  a similar fashion to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in analogy with Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10, the Hermite  Muttalib–Borodin ensembles exhibit a double-sided hard edge at the origin that is described  by the Meijer G-kernel (some discussion on this point is also given in [143]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us highlight the remarkable fact that applying the large eigenvalue approximation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) to our matrix product ensembles of interest does not disrupt the small eigenvalue  statistics enough to change the fact that their (double-sided) hard edge statistics are described  by the Meijer G-kernel — one need only reparametrise said kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, it is understood in  the present day that the matrix product ensembles reviewed in this section, together with  their approximations in terms of Muttalib–Borodin ensembles, belong to a universality class  of eigenvalue ensembles that are characterised by having their bulk, soft edge, and (double-  sided) hard edge statistics described respectively by the sine kernel, the Airy kernel, and  the Meijer G-kernel with various parametrisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This universality class has consequently  garnered great interest, with recent works showing that it contains other matrix product  ensembles [122], [216], [205], along with the class of Muttalib–Borodin ensembles with θ > 0  real and general weight of the form w(λ) = λae−NV(λ)χλ>0 [215], [247], [311] (see also [67],  [60] for related results on this latter class of Muttalib–Borodin ensembles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  57  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Integrals of Harish-Chandra–Itzykson–Zuber type  The Harish-Chandra–Itzykson–Zuber (HCIZ) integral formula is the evaluation  �  U(N) eTr(AUBU†) d ˜µ′  Haar(U) =  N−1  ∏  i=1  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Det[eajbk]N  j,k=1  ∆N(a)∆N(b) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)  where A and B are N × N complex Hermitian matrices with eigenvalues {aj}N  j=1 and {bj}N  j=1,  ∆N(λ) is the Vandermonde determinant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), and d ˜µ′  Haar(U) = dµ′  Haar(U)/vol(U(N)) is  the Haar probability measure on U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (The measure dµ′  Haar(U) is specified by equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) with β = 2, while vol(U(N)) = �  U(N) dµ′  Haar(U) can be obtained by setting β = 2 in  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) and multiplying the result by [vol(U(1))]N = [2π]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') The HCIZ integral  formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) is so named due to it being a special case of the general group integral  studied by Harish-Chandra in the 1957 work [175], and for its independent derivation and  introduction to random matrix theory by Itzykson and Zuber in the 1980 work [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The HCIZ integral and its numerous analogues (some of which can be inferred from  Harish-Chandra’s original group integral [175] as shown in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [150], [276]) have seen  direct applications in the study of quantum chromodynamics [203], [188], [9], [204] and have  otherwise been studied more broadly for their connections to Lie theory, combinatorics,  harmonic analysis, and probability theory [177], [295], [328].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In random matrix theory,  HCIZ-type integrals were shown in the early to mid-2000s to be powerful tools for studying  complex Gaussian sample covariance matrices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', products Σ−1W with W a complex  Wishart–Laguerre matrix, as specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, and Σ a fixed covariance matrix) [37],  [26], [288] and sums involving random matrices [194], [202], [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A breakthrough was made in the 2013 work [12] of Akemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', who showed how  the formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) can be used to obtain the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(cWm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) of  the complex Wishart product ensembles (recall Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) with arbitrary m ∈ N and  ν1 = · · · = νm = 0 (so that all of the Ginibre matrices G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Gm in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) are N × N)  — their method was subsequently extended in [11] to allow for arbitrary ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , νm > −1  through the use of induced Ginibre matrices (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the discussion below Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  success of these works in applying the HCIZ integral formula to arbitrarily large products  of random matrices prompted a series of studies on other random matrix products that  could likewise be treated using HCIZ-type integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, it was soon found that products  58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix Product Ensembles  of two coupled random matrices [10], [227] and products of arbitrarily many truncated  Haar-distributed unitary matrices [205] can be studied using appropriate generalisations of  the HCIZ integral formula (in fact, the authors of this latter work introduced a previously  unknown HCIZ-type integral formula as part of their development).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us now draw attention to an analogue and a generalisation of the HCIZ integral  formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) that are of particular interest to us: For N an even integer, the orthogonal  Harish-Chandra integral formula is the result  �  O(N) e  1  2 Tr(AUBUT) d ˜µ′  Haar(U) = 2−N/2  N/2−1  ∏  i=1  (2i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Det[2 cosh(ajbk)]N/2  j,k=1  ∏1⩽j<k⩽N/2(a2  k − a2  j )(b2  k − b2  j ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  where A and B are N × N real antisymmetric matrices with imaginary eigenvalues {±iaj}N/2  j=1  and {±ibj}N/2  j=1 , while d ˜µ′  Haar(U) = dµ′  Haar(U)/vol(O(N)) is the Haar probability measure  on O(N) (as before, set β = 1 in equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), and note that vol(O(1)) = 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Fixing M, N ∈ N such that 0 ⩽ M ⩽ N, letting η = diag(−IM, IN−M) be a pseudo-metric  tensor, and taking A and B to be N × N complex Hermitian matrices with eigenvalues  a1 < · · · < aM < 0 < aM+1 < · · · < aN,  b1 < · · · < bM < 0 < bM+1 < · · · < bN,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53)  the hyperbolic HCIZ integral satisfies the relation  �  U(η)/U(1)N e− Tr(AUBU−1) d ˜µHaar(U) ∝  Det[e−ajbk]M  j,k=1 Det[e−ajbk]N  j,k=M+1  ∆N(a)∆N(b)  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  where U(η) := {U ∈ GLN(C) | U†ηU = UηU† = η} is the pseudo-unitary group with  metric η and d ˜µHaar(U) is the Haar probability measure on the quotient space U(η)/U(1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) was made explicit in [276], using the fact that it is a special case of  Harish-Chandra’s group integral [175].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The hyperbolic HCIZ integral, on the other hand,  was introduced by Fyodorov [148], [150], who used the Duistermaat–Heckman localisation  principle to prove the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) is the canonical analogue of the HCIZ integral formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) for  the real case, so it stands to reason that it should be able to treat a matrix product ensemble  in the same manner as shown in [12], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One would be forgiven for thinking that the  real Wishart product ensembles could be studied using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52), but this is not the  59  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  case since the matrices A, B therein are antisymmetric (this being due to the fact that in  the general setting of [175], the matrices A, B must belong to the Lie algebra corresponding  to the Lie group that is being integrated over).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, it was shown in [11] that the real  Wishart product ensembles could be studied in the same way as in the complex case if one  knew a closed form expression, comparable to the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52), for  the left-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) with A, B real symmetric — such an integral is not of  HCIZ-type and is at best known to evaluate to a sum of ratios of zonal polynomials, rather  than anything resembling the right-hand side of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [310, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It turns out that the correct ensembles to consider are the antisymmetrised matrix product  ensembles introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11:  Inspired by Defosseux’s work [78] studying the antisymmetrised Laguerre ensemble  (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 with m = 1) and its quaternionic analogue through group theoretic methods,  Forrester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [144] used equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) to investigate the eigenvalue statistics of the  antisymmetrised matrix product ensembles, in effect deriving the results that we presented  earlier in Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It was likewise shown in [143] that the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(Hm)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) of the Hermitised matrix product ensemble of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 with  m ∈ N and ν1 = · · · = νm = 0 can be obtained via repeated applications of the relation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This marked the first instance of an HCIZ-type integral with non-compact domain  of integration being successfully applied to the study of matrix product ensembles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the key  distinction to previous studies was that the constraints (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) now needed to be taken into  account, which are necessary for the integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54) to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  Outline of the Thesis  Thus far in this chapter, we have introduced the random matrix ensembles and the statistics of  those ensembles that are at the focus of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Namely, we are interested in the classical  matrix ensembles and matrix product ensembles reviewed in Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  More precisely, we will be studying the eigenvalue densities of these ensembles, along with  their (mixed) moments and cumulants and the generating functions of these moments and  cumulants, which are all defined in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned at the beginning of this chapter, our  studies will in some sense culminate with the derivation of two types of recursions: 1-point  60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Outline of the Thesis  recursions for the series coefficients Mk,l (see Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) of the classical matrix ensembles’  spectral moments and loop equations for the connected n-point correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) of global scalings of the (N, N) Hermitised and antisymmetrised Laguerre ensembles  of Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loosely speaking, our motivation for studying  these two classes of random matrix ensembles together is that they both relate to the Ginibre  ensembles of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, while our decision to study 1-point recursions alongside loop  equations stems from the deeper desire to understand the connection, if any, between  these two type of recursive schemes — such a connection can be anticipated and has been  conjectured [61] to exist due to the facts that these recursions characterise closely related  statistics and that ensembles known to admit 1-point recursions are often governed by loop  equations, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In addition to the 1-point recursions and loop equations derived in this  thesis, we also review and develop some theory on differential equations and ribbon graph  combinatorics, along with some exotic topological hypermaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We commence Chapter 2 with a review of some theory on Selberg correlation integrals  (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) and some relations between the Cauchy and Jacobi ensembles (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The theory of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 is then used in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 to derive third order (β = 2) and fifth  order (β = 1, 4) linear differential equations for the eigenvalue densities ρ(J)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) and  resolvents W(J)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) of the Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The theory of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and limiting  procedures drawn from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 are subsequently used in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 to obtain  analogous differential equations for the Laguerre and Cauchy ensembles — we also present  the intermediate differential equations for the shifted Jacobi ensembles corresponding to the  weight w(sJ)(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 concludes with §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, where an alternate extension of  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 is used to derive seventh order linear differential equations for the eigenvalue  densities and resolvents of the β = 2/3, 6 Gaussian β ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This serves to demonstrate  the fact that our method is technically applicable to any classical β ensemble with either  β ∈ 2N or 4/β ∈ 2N (albeit with difficulty growing rapidly for each new member of these  sequences of ensembles in their natural order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The primary contribution of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 is the exposition of how Selberg correlation  integral theory provides a unified approach for obtaining the aforementioned differential  equations characterising eigenvalue densities and resolvents of classical β ensembles with  β ∈ {σ ∈ R | σ ∈ 2N or 4/σ ∈ 2N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, many of these differential equations have been  61  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  derived earlier in the literature through a variety of methods: The differential equation for  the GUE has been given in [221], [164];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for the GOE and GSE in [319];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for a special case of  the shifted JUE (sJUE) in [151];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for the LUE in [164], [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and for the symmetric (α ∈ R) CyUE  in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We confirm that our method recovers these differential equations and additionally  derive new analogues for the (un-shifted) Jacobi ensembles, the LOE and LSE, the symmetric  (a = b) shifted Jacobi ensembles, the symmetric CyOE and CySE, the non-symmetric (α ∈ C)  CyUE, and the Gaussian β ensembles with β = 2/3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The differential equations of [164]  were used therein to obtain optimal bounds for the rate of convergence to the limiting  semi-circle law ρ(G),0(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) (for the GUE) and Marˇcenko–Pastur law ρ(L),0(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  (for the LUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More recently, Kopelevitch [213] used the third order differential equation  for the GUE eigenvalue density to study the convergence properties of 1/N expansions for  averages of linear statistics of the GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, we modify the results of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 to  give differential equation characterisations of the classical matrix ensembles in the global and  edge scaling regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These characterisations complement a plethora of existing results on  statistics of the classical matrix ensembles in said scaling regimes, such as the Tracy–Widom  law [299], [300], the characterisations of the classical unitary ensembles in terms of the  sine, Bessel, and Airy kernels [115], [39], [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 7], the loop equations for the classical β  ensembles [319], [142], and much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Chapter 3 is concerned with the (mixed) moments mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) and cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) of the ensembles introduced in Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As a second application of the  contents of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we use the differential equations therein to derive linear recurrence  relations on the spectral moments mk of the classical matrix ensembles, which we present  in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be precise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we give recurrences on the spectral moments m(J)  k  and m(L)  k  of the  Jacobi and Laguerre ensembles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the differences of even moments µ(sJ)  k  = m(sJ)  2k+2 − m(sJ)  2k  of  the symmetric shifted Jacobi ensembles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the sums of even moments µ(Cy)  k  = m(Cy)  2k+2 + m(Cy)  2k  of the symmetric Cauchy ensembles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and the spectral moments m(G)  k  of the Gaussian β  ensembles with β = 2/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 6 (we also confirm that our method recovers the GUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' GOE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  and GSE recurrences of [174],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [224]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Of these recurrences, those pertaining to the unitary  ensembles have been given earlier in [169], [223], [71], [72], [23] — the work [71] also contains  inhomogeneous analogues of our LOE and LSE moment recurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A benefit of our  approach is that we are able to show that the moment recurrences of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 are valid for  62  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Outline of the Thesis  complex k (so long as the involved moments mk are convergent), thereby generalising the  LUE and sJUE moment recurrences given in [169], [223], [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we substitute the  moment expansions of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 into the recurrences of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 to extract 1-point recursions  for the moment expansion coefficients Mk,l associated with the Laguerre ensembles, the  Gaussian β ensembles with β = 2/3, 6, and the (shifted) Jacobi unitary ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' That is, we  show that particular finite linear combinations of the Mk,l, whose coefficients are polynomials  in k, are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Examples of such 1-point recursions include the celebrated Harer–  Zagier recursion [174] for the M(GUE)  k,l  and the analogous recursions for the GOE and GSE  given by Ledoux [224] — we confirm that our method recovers these recursions and we also  find that our LUE recursion is a natural extension of [86, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1] to the a ̸= 0 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The analysis of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 complements a multitude of studies on the spectral moments  of the classical matrix ensembles, with many of these works highlighting connections to skew-  orthogonal polynomial theory [169], [223], [224], [229], [240], [71], symmetric function theory  [199], [27], [92], [96], [267], [149], [242], and hypergeometric orthogonal polynomials from the  Askey scheme [72], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For the sake of completeness, we survey these works in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In addition to the theory reviewed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, a major motivation for establishing 1-point  recursions on moment expansion coefficients Mk,l of the classical matrix ensembles is that,  in the Gaussian and Laguerre cases, said coefficients are well known to enumerate certain  types of ribbon graphs and (hyper)maps [36], [329], [82], [57], [219] — these combinatorial  objects have in turn seen applications in algebraic geometry and mathematical physics  [83], [250], [220], [248], [304], [53], [287], [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we begin Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 by giving a  pedagogically extensive review of how the Isserlis–Wick theorem [186], [312] can be used to  prove the aforementioned connection between ribbon graphs and the spectral moments of  the Gaussian (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) and Laguerre (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, we go further and elucidate  the relationship between ribbon graphs and the mixed moments and cumulants of said  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This theory is then extended in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 to give combinatorial interpretations for the  mixed cumulants of the Hermitised and antisymmetrised matrix product ensembles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in short,  we show that these cumulants enumerate ribbon graphs that can be seen as amalgamations  of the ribbon graphs discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In Chapter 4, we transition to an analytic study of the Hermitised and antisymmetrised  Laguerre ensembles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the matrix product ensembles of Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 with  63  CHAPTER 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Introduction  m = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that our matrix product ensembles of interest are not (immediately)  amenable to the techniques of Chapter 2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 due to the relative intractibility (as  compared to the orthogonal polynomials relevant to the classical matrix ensembles) of the  Meijer G-functions displayed in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we take this opportunity to showcase  the loop equation formalism, which is well suited to studying the relevant matrix products  H1 = G†  1HG1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) and J1 = GT  1 JN0G1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) due to their being constructed from Ginibre  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Aside from shedding light on the eigenvalue statistics, recently studied in [143],  [144], of the Hermitised and antisymmetrised matrix product ensembles, our development  provides an avenue for exploring properties of loop equations for matrix product ensembles  in general (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we give a brief review of loop equations for the  classical β ensembles and the closely related theory of the topological recursion, before  moving onto Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, where we derive loop equations for the antisymmetrised  and Hermitised Laguerre ensembles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we first derive  loop equations for the unconnected n-point correlators Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) associated with  appropriate global scalings of the antisymmetrised and Hermitised Laguerre ensembles,  use those loop equations and equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) to obtain loop equations for the connected  analogues Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn), and then finally show how inserting the large N expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  into the second set of loop equations enables us to compute the expansion coefficients  Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) in a systematic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We round out Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 with some concrete  computations of W0  1(x1) and W0  2(x1, x2), which allow us to perform some consistency checks  against the combinatorics of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 contains a brief discussion on some  open questions that proved to be outside the scope of this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  64  Chapter 2  Differential Equations for the  Classical Matrix Ensembles  The primary goal of this chapter is to establish linear differential equations satisfied by  the eigenvalue densities and resolvents of the classical matrix ensembles introduced in  Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7), replacing N by N + 1  in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) shows that the eigenvalue density is given in terms of the absolute value  of the βth moment of the characteristic polynomial according to  ρ(w)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β) =  N + 1  N (w)  N+1,β  �  RN  N+1  ∏  i=1  w(λi) |∆N+1(λ)|β dλ1 · · · dλN  ������  λN+1=λ  =  (N + 1)N (w)  N,β  N (w)  N+1,β  w(λ)  �  N  ∏  i=1  |λ − λi|β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  The average here is over the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) for N eigenvalues, in  keeping with equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The crux of this chapter is that when w(λ) is taken to be the  Jacobi weight, this average is an example of a particular class of Selberg correlation integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We review these integrals in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, along with some related theory that is needed to  obtain the sought linear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we use the contents of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 to establish order three (β = 2) and order  five (β = 1, 4) linear differential equations that characterise the eigenvalue densities ρ(J)(λ)  and resolvents W(J)  1 (x) of the Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From them we derive analogous differential  65  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  equations for the other classical ensembles in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Cauchy case, these follow  from the change of variables suggested by equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) and some subtleties that  we outline in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Laguerre case, we simply apply some limiting procedures  drawn from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 to the differential equations of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar limiting procedures are  applicable in the Gaussian case as well, though the corresponding differential equations are  already well known [221], [164], [319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, we instead use Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 to tweak the proofs  given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and treat the Gaussian β ensembles with β = 6 and, through the β ↔ 4/β  duality relation displayed in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, β = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We choose to study these Gaussian β  ensembles because they are the next easiest classical β ensemble to address — while our  techniques are valid for all classical weights and β = σ, 4/σ with σ a positive even integer,  there is a technical increase in complexity as the weight changes from Gaussian, to Laguerre,  to Jacobi, and also when σ is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For the sake of completeness and to illustrate the expected form of the results in Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we present the GOE, GUE, and GSE differential equations here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We use [319] as our  source and thus take the Gaussian weight to be w(g)(λ) := e−βNλ2/(4g), noting this change  of weight with the superscript (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (The coupling constant g determines the length scale:  To leading order in N, the spectrum is supported on (−2√g, 2√g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The weight w(g)(λ) is  equivalent to the weight w(G)(λ) specified in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) upon setting g = βN/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define  D(g)  N,β =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  g  √κN  �2 d3  dx3 − y2  (g)  d  dx + x,  β = 2,  −  �  g  √κN  �4 d5  dx5 + 5  �  1  2y2  (g) − h  �  g  √κN  �� �  g  √κN  �2 d3  dx3 − 3  �  g  √κN  �2  x d2  dx2  −  �  y4  (g) − 4h  �  g  √κN  �  y2  (g) −  �  g  √κN  �2�  d  dx +  �  y2  (g) − 2h  �  g  √κN  ��  x,  β = 1, 4,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  where κ := β/2, h := √κ − 1/√κ, and y(g) :=  �  x2 − 4g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for β = 1, 2, and 4,  D(g)  N,β ρ(g)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  and  D(g)  N,β  1  NW(g)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) =  �  �  �  �  �  2,  β = 2,  2y2  (g) − 10h  �  g  √κN  �  ,  β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  66  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It can be observed that for β = 1 or 4,  D(g)  N,β = −  �  g  √κN  �2 �  D(g)  N,2 + 2x  � d2  dx2 +  �  y2  (g) − 5h  �  g  √κN  ��  D(g)  N,2  + 1  2y2  (g)  �  g  √κN  �2 d3  dx3 −  �  hy2  (g) −  �  g  √κN  �� �  g  √κN  � d  dx + 3h  �  g  √κN  �  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  This form of the differential operator should be compared with differential equations  for the GOE and GSE eigenvalue densities that were derived recently in [260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  differential equations of [260] are linear, but differ from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) in that they are third  order and have inhomogeneous terms dependent on the GUE eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The operator for β = 2 is even in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case of β = 1 and 4, it is invariant under the  mapping (N, κ) �→ (−Nκ, 1/κ), which is consistent with the known duality relations  presented in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, the corresponding duality relations  for the Laguerre and Jacobi ensembles given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 are what allow us to obtain  differential equations for β = 1 in the upcoming derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They also suggest that in  certain cases, the LUE and JUE spectral moments mk, when scaled to be O(N), are odd  functions in N (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The structure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) tells us that fN+1,β(x) := ρ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β)/w(x) has a large x expan-  sion of the form  fN+1,β(x) x→∞  =  ∞  ∑  k=0  akxβN−k  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  with a0 = (N + 1)NN,β/NN+1,β fixed and all other constant coefficients ak yet to be  determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, for β an even integer, this series must terminate since  the average in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) is manifestly a polynomial of degree βN in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting  ρ(g)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) = w(g)(x) f (g)  N,2(x) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) shows (choosing the GUE for simplicity),  �� g  N  �2 d3  dx3 − 3g  N x d2  dx2 +  �  2x2 + g  N (4N − 3)  � d  dx − 4(N − 1)x  �  f (g)  N,2(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  One can check that this differential equation admits a unique polynomial solution  consistent with equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), and with {ak}k>0 completely determined by the choice  of a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This latter feature is true of all the differential equation characterisations we will  obtain for ρ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It holds true because the underlying differential-  difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) is effectively a (multi-dimensional) first order recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  67  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  The third order differential equation for the GUE eigenvalue density (equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  with β = 2) can be traced back to the work of Lawes and March [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There, the setting is  that of interpreting the GUE eigenvalue density as the squared ground state wave function of  spinless non-interacting fermions in one dimension, in the presence of a harmonic confining  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the random matrix theory literature, this result appeared in the work of G¨otze  and Tikhomirov [164], making use of an earlier result of Haagerup and Thorbjørnsen [169]  characterising the two-sided Laplace transform of the density in terms of a hypergeometric  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) with β = 1, 4 were derived in [319] using  a method based on known evaluations of the eigenvalue densities in terms of Hermite  polynomials (recall the discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, particularly that which concerns [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our  approach, using Selberg correlation integrals and duality formulae, is different to those just  recounted, and moreover unifies all the classical cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Selberg Correlation Integrals  As mentioned above, our initial interest lies in the Selberg correlation integral that is specified  as the average of ∏N  i=1 |λ − λi|β (0 < λ < 1, β > 0) against the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the Jacobi β ensemble,  which we recall from equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) as  p(J)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (J)  N,β  N  ∏  i=1  λa  i (1 − λi)b |∆N(λ)|β,  0 < λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN < 1, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  The normalisation constant N (J)  N,β is known as the Selberg integral, which, in turn, explains  why the average (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) is referred to as a Selberg correlation integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  The Selberg and Dixon–Anderson integrals  Highlighting the dependence on the parameters a, b, we denote the Selberg integral by  SN(a, b, κ) := N (J)  N,β =  �  [0,1]N  N  ∏  i=1  λa  i (1 − λi)b |∆N(λ)|2κ dλ1 · · · dλN,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  where we now allow a, b ∈ C with Re(a), Re(b) > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This integral was first computed by  Selberg [285] to be given by  SN(a, b, κ) =  N−1  ∏  j=0  Γ(a + 1 + κj)Γ(b + 1 + κj)Γ(1 + κ(j + 1))  Γ(a + b + 2 + κ(N + j − 1))Γ(1 + κ)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  68  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Selberg Correlation Integrals  with the requirements Re(a), Re(b) > −1 now seen to be necessary for avoiding the poles in  the numerator of the right-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since the work of Selberg, there  have been a number of proofs produced for the above evaluation [88], [22], [20] (see [119,  Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To make connection with the recursive theme of this thesis, we present  below the method of Anderson, which consists of setting up a recurrence in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Following [119, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2], we fix parameters s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , sN+1 > 0 and refer to the quantity  D(s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',sN+1)  N  (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , aN+1) :=  �  aN+1<λN<···<λ1<a1  N  ∏  i=1  N+1  ∏  j=1  |λi − aj|sj−1 ∆N(λ) dλ1 · · · dλN  = Γ(s1) · · · Γ(sN+1)  Γ(s1 + · · · + sN+1)  ∏  1⩽j<k⩽N+1  (aj − ak)sj+sk−1  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  as the Dixon–Anderson integral, so-called because of its independent derivations in 1905 by  Dixon [85] and in 1991 by Anderson [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The equality of both sides of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) is  proven in [119] as follows: Let {wi}N+1  i=1 be drawn from the Dirichlet distribution, which we  specify by the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Γ(s1 + · · · + sN+1)  Γ(s1) · · · Γ(sN+1)  N+1  ∏  i=1  wsi−1  i  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  supported on the standard N-simplex  �  (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , wN+1) ∈ RN+1 ���  N+1  ∑  i=1  wi = 1 and wi > 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, introduce the random rational function  R(λ) :=  N+1  ∑  i=1  wi  ai − λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  In a similar fashion to the calculation surrounding the ratio of characteristic polynomials  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='87) of the matrix model MN (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='86) for the Gaussian β ensemble that was outlined in  §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, taking the residues of both sides of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) at λ = ai expresses the wi in  terms of the {ai}N+1  i=1 and the zeroes {λi}N  i=1 of R(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting these expressions into the  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) and including the Jacobian yields the Dixon–Anderson j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',sN+1)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , aN+1) := Γ(s1 + · · · + sN+1)  Γ(s1) · · · Γ(sN+1)  N  ∏  i=1  N+1  ∏  p=1  |λi − ap|sp−1  ×  ∏  1⩽j<k⩽N  (aj − ak)1−sj−sk ∆N(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  69  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  A graphical argument reveals the interlacing condition aN+1 < λN < · · · < λ1 < a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  integrating ps1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',sN+1(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , aN+1) in the variables λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN over this region  gives equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To prove the Selberg integral formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), consider the generalised integral [119,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2]  KN(a, b, κ) :=  �  Ω  N+1  ∏  l=1  xa  l (1 − xl)b  N  ∏  i=1  N+1  ∏  j=1  |yi − xj|κ−1  × |∆N(y)∆N+1(x)| dx1 · · · dxN+1dy1 · · · dyN,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  where Ω denotes the region 0 < xN+1 < yN < · · · < y1 < x1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Evaluating this integral  over the {xi}N+1  i=1 first reveals that  KN(a, b, κ) =  �  0<yN<···<y1<1 D(b+1,(κ)N,a+1)  N+1  (1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , yN, 0) |∆N(y)| dy1 · · · dyN  = SN(a + κ, b + κ, κ)Γ(a + 1)Γ(b + 1)Γ(κ)N  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='Γ(a + b + 2 + Nκ) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  where (κ)N denotes the N-tuple (κ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , κ) and D(b+1,(κ)N,a+1)  N+1  is the Dixon–Anderson integral  as defined in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The second line here follows from substituting in the right-  hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), recalling the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) of the Selberg integral, and  introducing a factor of N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' to symmetrise the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If one instead evaluates KN(a, b, κ) by  integrating over the {yi}N  i=1 first, one observes that  KN(a, b, κ) =  �  0<xN+1<···<x1<1 D(κ)N+1  N  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN+1) |∆N+1(x)| dx1 · · · dxN+1  = SN+1(a, b, κ)  Γ(κ)N+1  (N + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='Γ((N + 1)κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  Comparing the two expressions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) for KN(a, b, κ) results in the recurrence  SN+1(a, b, κ) = SN(a + κ, b + κ, κ)(N + 1)Γ((N + 1)κ)Γ(a + 1)Γ(b + 1)  Γ(a + b + 2 + Nκ)Γ(κ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  Solving this recurrence via induction on N, we finally obtain equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Normalisation constants for the classical β ensembles  By the definition of the Selberg integral, the normalisation constant N (J)  N,β in the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the Jacobi β ensemble is given by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain the corresponding  70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Selberg Correlation Integrals  normalisation constant for the Laguerre β ensemble, we extend Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 by setting  λi = γi/b in the right-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) and taking the limit b → ∞ to see that  N (L)  N,β :=  �  [0,∞)N  N  ∏  i=1  λa  i e−λi |∆N(λ)|2κ dλ1 · · · dλN  = lim  b→∞ bN(κ(N−1)+a+1)SN(a, b, κ)  =  N−1  ∏  j=0  Γ(1 + κ(j + 1))Γ(a + 1 + κj)  Γ(1 + κ)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  where the final line is obtained by incorporating Selberg’s evaluation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and using  Stirling’s approximation (recall that κ := β/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the Gaussian case, one similarly sets λi = (1 + γi/  √  L)/2 and a = b = L in the  right-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), so that taking the limit L → ∞ shows that  N (G)  N,β :=  �  RN  N  ∏  i=1  e−λ2  i |∆N(λ)|2κ dλ1 · · · dλN  = lim  L→∞ 2−N(2L+κ(N−1)+1)L−N(κ(N−1)+1)/2SN(L, L, κ)  = (2π)N/22−N(κ(N−1)+1)/2  N−1  ∏  j=0  Γ(1 + κ(j + 1))  Γ(1 + κ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  The normalisation constant for the Cauchy β ensemble is given by  N (Cy)  N,β :=  �  RN  N  ∏  i=1  (1 − iλ)η(1 + iλ)η |∆N(λ)|2κ dλ1 · · · dλN  = 2−N(κ(N−1)+2Re(α))πN  N−1  ∏  j=0  Γ(2Re(α) + 1 + κj)Γ(1 + κ(j + 1))  Γ(α + 1 + κj)Γ(α + 1 + κj)Γ(1 + κ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  where we have retained our choice of reparametrisation η = −κ(N − 1) − 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The above  evaluation can be obtained by changing variables in the Selberg integral, but the necessary  analytic continuation is a little subtle, so we defer proof of this fact to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We conclude this subsection by noting that Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 is now complete (so too is the  definition of the classical β ensembles given in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), with the normalisation constants of  the classical matrix (β) ensembles’ eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) specified by equations  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From the discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, substituting these values into  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) (using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), as well) then completes Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 by  making the partition functions ZN explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  71  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Further preliminaries concerning Selberg correlation integrals  As outlined thus far, our initial object of interest is the average displayed in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  with w(λ) = λa(1 − λ)b the Jacobi weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For later reference, we denote it  I(J)  N,β(λ) :=  �  N  ∏  i=1  |λ − λi|β  �  JEN,β(a,b)  =  N (J)  N+1,β ρ(J)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β)  (N + 1)N (J)  N,β w(J)(λ)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  where the subscript JEN,β(a, b) indicates that the average is with respect to the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(J)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) made explicit in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In order to derive differential  equations satisfied by the corresponding eigenvalue density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), we must make a detour  and first study the auxiliary average  J(N)  n,p (λ) :=  �  n  ∏  i=1  (λi − λ)N+χi⩽p  �  JEn,4/β(a′,b′)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  The significance of J(N)  n,p (λ) is that it satisfies the differential-difference equation [114]  (n − p)EpJ(N)  n,p+1(λ) = −(Apλ + Bp)J(N)  n,p (λ) + λ(λ − 1) d  dλ J(N)  n,p (λ) + Dpλ(λ − 1)J(N)  n,p−1(λ),  Ap = (n − p)  �  a′ + b′ + 2  κ(n − p − 1) + 2N + 2  �  ,  Bp = (p − n)  �  a′ + N + 1 + 1  κ(n − p − 1)  �  ,  Dp = p  � 1  κ(n − p) + N + 1  �  ,  Ep = a′ + b′ + 1  κ(2n − p − 2) + N + 2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  later observed to be equivalent to a particular matrix differential equation [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The order p elementary symmetric polynomial on N variables is  ep(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) =  ∑  1⩽j1<j2<···<jp⩽N  xj1 · · · xjp  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  with the convention ep(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) = 0 if p > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is known [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4] that the above  differential-difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) is satisfied by a broader class of functions  ˜J(N)  n,p,q(λ) :=  1  NN,p  �  n  ∏  i=1  |λi − λ|N χλ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',λN−q<x ep(λ − λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λ − λN)  �  JEn,4/β(a′,b′)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  where NN,p := (N  p), this being the number of terms in the sum on the right-hand side of  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case q = N, we see that ˜J(N)  n,p,q(λ) is equal to J(N)  n,p (λ) due to symmetry  of the integrand in the right-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  72  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Selberg Correlation Integrals  The differential-difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) can be seen to be equivalent to a closed system  of differential equations upon letting p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, appropriately substituting these  n + 1 equations into each other (more details in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) yields an order n + 1 linear differential  equation for J(N)  n,0 (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Making the substitutions  (n, N, β, a′, b′) �→ (N, β, 4/β, a, b)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), with β a positive even integer to remove the absolute value sign, then  reveals that the aforementioned differential equation becomes order N + 1 and is satisfied  by I(J)  N,β(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, even though our present goal is to obtain a linear differential  equation for I(J)  N,β(λ), a differential equation of order N + 1 is not suitable for the large N  analysis that we wish to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, an alternative approach must be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Instead of making the substitutions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20), we opt to use the duality [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 13]  SN(a, b, κ)  SN(a + n, b, κ)  �  N  ∏  i=1  (λi − λ)n  �  JEN,β(a,b)  =  �  n  ∏  i=1  (1 − λλi)N  �  JEn,4/β(a′,b′)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  where SN is the Selberg integral as defined in the previous subsection, and we now take  a′ = 1  κ (a + b + 2) + N − 2,  b′ = −1  κ (b + n) − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  This duality relation can be drawn from the fact that both sides of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) are  known to equal [200]  2F(κ)  1  (−N, (a + b + n + 1)/κ + N − 1, (a + n)/κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (x)n) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  where 2F(κ)  1  is the generalised multivariate hypergeometric function and (x)n is again the  n-tuple (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that on the right-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), the parameter  b′ is in general less than −1, and so the integral must be understood in the sense of  analytic continuation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 forthcoming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The left-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) is  proportional to I(J)  N,β(λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) upon taking n = β a positive even integer, while a simple  change of variables shows that the right-hand side is equal to (−λ)nNJ(N)  n,0 (1/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Combining  these two facts, we thus have for β a positive even integer that  I(J)  N,β(λ) ∝ (−λ)βN J(N)  β,0 (1/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  Hence, the order n + 1 differential equation for J(N)  n,0 (x) discussed in the paragraph prior trans-  lates to an order β + 1 linear differential equation for I(J)  N,β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These differential equations  are derived in detail in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  73  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Relating the Cauchy and Jacobi Ensembles Through Analytic  Continuation  At the end of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we said that in deriving results for the Cauchy ensembles, we will go  through the shifted Jacobi ensembles, which correspond to the weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  w(sJ)(λ) = (1 − λ)a(1 + λ)bχ−1<λ<1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In doing so, not only will we bring more clarity to our derivations, but we will also have  the benefit of making our results explicit in the Jacobi case with the shifted weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  latter point is mainly one of convenience, both for the reader interested in the shifted Jacobi  ensembles, and ourselves when comparing against the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Relating the  un-shifted Jacobi ensembles to their shifted counterparts is very easy: One need only make  the change of variables λi �→ 1  2(1 − λi), where {λi}N  i=1 are the eigenvalues — there are no  complications in applying this change of variables in the upcoming calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Making  this change of variables in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), we see that the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the shifted  Jacobi β ensemble is  p(sJ)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (sJ)  N,β  N  ∏  i=1  (1 − λi)a(1 + λi)b |∆N(λ)|β,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  where −1 < λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN < 1, β = 2κ > 0, and  N (sJ)  N,β = 2N(κ(N−1)+a+b+1)N (J)  N,β = 2N(κ(N−1)+a+b+1)SN(a, b, κ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  with the value of SN(a, b, κ) given in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Denoting an average with respect to  this j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' by the subscript sJEN,β(a, b), in keeping with the notation in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15),  the spectral moments (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) of the shifted and un-shifted Jacobi β ensembles are related as  follows:  m(sJ)  k  =  �  N  ∑  i=1  λk  i  �  sJEN,β(a,b)  =  �  N  ∑  i=1  (1 − 2λi)k  �  JEN,β(a,b)  =  k  ∑  s=0  �k  s  �  (−2)s  �  N  ∑  i=1  λs  i  �  JEN,β(a,b)  =  k  ∑  s=0  �k  s  �  (−2)sm(J)  s ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  where the binomial expansion has been used in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  74  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Relating the Cauchy and Jacobi Ensembles Through Analytic Continuation  Having established the mechanisms for moving between shifted and un-shifted Jacobi  ensembles, the rest of this section is devoted to relating the Cauchy ensembles to the shifted  Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We first discuss the symmetric case (recall from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that this corresponds  to taking a = b and α ∈ R), where the necessary arguments are easier to follow, before  outlining the (non-symmetric) general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Relating the symmetric Cauchy and shifted Jacobi ensembles  When a = b and α ∈ R, the Cauchy and shifted Jacobi weights w(Cy)(λ) = (1 + λ2)η and  w(sJ)(λ) = (1 − λ2)aχ−1<λ<1 are even functions of λ (if α := −κ(N − 1) − 1 − η is real, then  so too is η), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', symmetric about λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Integrations of multivariable symmetric functions  against these weights can be related via the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) be a multivariable symmetric polynomial of degree d in each xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For 2η < −(d + 1), define  I(Cy)  N,η [ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] :=  � ∞  −∞ dx1 (1 + x2  1)η · · ·  � ∞  −∞ dxN (1 + x2  N)η f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  and for η outside of this range, define I(Cy)  N,η [ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] by its analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Also, in  relation to the shifted Jacobi weight with a = b > −1, define  I(sJ)  N,a [ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] :=  � 1  −1 dx1 (1 − x2  1)a · · ·  � 1  −1 dxN (1 − x2  N)a f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  and for a outside of this range, define I(J)  N,a[ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] by its analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have  I(Cy)  N,η [ f (ix1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ixN)] = (tan πη)NI(sJ)  N,η [ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' xN)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let p ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Suppose 2η < −(p + 1) and a > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A simple change of variables (for  k even) and use of the Euler beta integral evaluation (see [119, Exercises 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2]) shows that  � ∞  −∞(1 + x2)ηxk dx =  �  �  �  �  �  0,  k odd,  (−1)k/2 tan πη Γ(1+η)Γ((k+1)/2)  Γ((k+3)/2+η)  ,  k even,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  and  � 1  −1(1 − x2)axk dx =  �  �  �  �  �  0,  k odd,  Γ(1+a)Γ((k+1)/2)  Γ((k+3)/2+a)  ,  k even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  75  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  The functions f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) and f (ix1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ixN) are polynomials, so the computation of  I(Cy)  N,η [ f (ix1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ixN)] and I(sJ)  N,a [ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] reduces to the above one-dimensional integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since, as analytic functions of η, we read off from the respective evaluations that  � ∞  −∞(1 + x2)η(ix)2k dx = tan πη  � 1  −1(1 − x2)ηx2k dx,  the stated result (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) follows  One immediate consequence is a proof for the value of N (Cy)  N,β given in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14),  in the symmetric case with κ = β/2 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let α be real and let η = −κ(N − 1) − 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β even, we have  (−1)κN(N−1)/2N (Cy)  N,β = (− tan πα)NN (sJ)  N,β  ���  a=b=−κ(N−1)−1−α,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  where both sides are to be interpreted as analytic functions in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With α and thus η real, we have  N (Cy)  N,β =  � ∞  −∞ dx1(1 + x2  1)η · · ·  � ∞  −∞ dxN(1 + x2  N)η  ∏  1⩽j<k⩽N  |xk − xj|β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Also, with a = b, we have  N (sJ)  N,β =  � 1  −1 dx1 (1 − x2  1)a · · ·  � 1  −1 dxN (1 − x2  N)a  ∏  1⩽j<k⩽N  |xk − xj|β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For β even, ∏1⩽j<k⩽N |xk − xj|β is a multivariable symmetric polynomial, so application of  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 gives equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The explicit form of the analytic continuations in α of both sides of equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) is known from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the notation therein, the  equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) requires  SN(−κ(N − 1) − 1 − α, −κ(N − 1) − 1 − α, κ)  = (−1)κN(N−1)/2(−2π cot πα)N  N−1  ∏  j=0  Γ(2α + 1 + κj)Γ(1 + κ(j + 1))  Γ(α + 1 + κj)2Γ(1 + κ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  Under the assumption that β is even, this can be checked upon the manipulation j �→  N − 1 − j in the product defining SN, and then use of the reflection equation for the gamma  functions in that product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Agreement with the right-hand side of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  76  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Relating the Cauchy and Jacobi Ensembles Through Analytic Continuation  We can make use of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and further apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to relate the eigenvalue  densities of the Cauchy and shifted Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the setting of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1,  ρ(Cy)(iλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = − cot πα ρ(sJ)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)  ���  a=b=−κ(N−1)−1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A straightforward extension of the above proposition is the relation  ρ(Cy)  k  (iλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , iλk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = (− cot πα)kρ(sJ)  k  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)  ���  a=b=−κ(N−1)−1−α  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  on the k-point correlation functions defined by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A similar such extension  holds true for Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 forthcoming, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Although established for β an even integer via reasoning based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2,  an application of Carlson’s theorem to the differences of the left- and right-hand sides of  equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) interpreted as functions of κ − 1 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [119, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1]) shows  that Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 remain true for general β > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Carlson’s theorem states  that if a function f (κ − 1) is analytic for Re(κ) ⩾ 1, has the bound | f (κ − 1)| = O(eτ|κ−1|)  for some τ < π, and vanishes at all κ ∈ N, then f ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For the particular values of β even, β = 2, 4, we will use the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) relating the  eigenvalue density of the Cauchy ensemble for α real to (an analytic continuation of) the  eigenvalue density of the shifted Jacobi ensemble supported on (−1, 1) with a = b and the  same value of β to study properties of the former — extension to β = 1 is enabled through  the above remark or Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is presented in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, but first we outline a relationship  between the Cauchy ensemble when Im(α) ̸= 0 (also known as the non-symmetric case or  the generalised Cauchy ensemble) and the shifted Jacobi ensemble now requiring a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Relating the non-symmetric Cauchy and shifted Jacobi ensembles  A classical result of Cauchy [59] gives  � ∞  −∞  dt  (1 − it)γ(1 + it)δ = 22−γ−δπ Γ(γ + δ − 1)  Γ(γ)Γ(δ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  subject to the requirement that Re(γ + δ) > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' outside of this range we consider the integral  as defined by the analytic continuation given by the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Use of the reflection  77  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  equation for the gamma function allows this to be rewritten  � ∞  −∞(1 − it)γ(1 + it)δ dt = 2γ+δ+2 sin πγ sin πδ  sin π(γ + δ)  Γ(γ + 1)Γ(δ + 1)  Γ(γ + δ + 2)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  subject now to the requirement Re(γ + δ) < −1 on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) is to be compared against the Euler beta function evaluation  � 1  0 tc(1 − t)d dt = Γ(c + 1)Γ(d + 1)  Γ(c + d + 2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  or, equivalently,  � 1  −1(1 − t)c(1 + t)d dt = 2c+d+1 Γ(c + 1)Γ(d + 1)  Γ(c + d + 2)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  where on the left-hand side, it is required that Re(c), Re(d) > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The agreement in the  gamma function dependence of both integrals allows for a relation between multiple integrals  analogous to that in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) be a multivariable symmetric polynomial of degree ˜d in each xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For Re(γ + δ) < − ˜d − 1, define  ˜I(Cy)  N,γ,δ[ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] :=  � ∞  −∞ dx1 (1 − ix1)γ(1 + ix1)δ · · ·  · ·  � ∞  −∞ dxN (1 − ixN)γ(1 + ixN)δ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  and for γ, δ outside of this range, define ˜I(Cy)  N,γ,δ[ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] by its analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Also, in  relation to the shifted Jacobi weight with Re(c), Re(d) > −1, define  ˜I(sJ)  N,c,d[ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] :=  � 1  −1 dx1 (1 − x1)c(1 + x1)d · · ·  · ·  � 1  −1 dxN (1 − xN)c(1 + xN)d f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  and for c, d outside of this range, define ˜I(sJ)  N,c,d[ f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN)] by its analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have  ˜I(Cy)  N,γ,δ[ f (1 − ix1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 1 − ixN)] =  �  2sin πγ sin πδ  sin π(γ + δ)  �N  ˜I(sJ)  N,γ,δ[ f (1 − x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1 − xN)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For p ∈ Z≥0, it follows from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) upon setting c = γ and  d = δ that  � ∞  −∞(1 − it)γ+p(1 + it)δ dt = 2γ+δ+p+2 sin πγ sin πδ  sin π(γ + δ)  Γ(γ + p + 1)Γ(δ + 1)  Γ(γ + δ + p + 2)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  78  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Relating the Cauchy and Jacobi Ensembles Through Analytic Continuation  and  � 1  −1(1 − t)γ+p(1 + t)δ dt = 2γ+δ+p+1 Γ(γ + p + 1)Γ(δ + 1)  Γ(γ + δ + p + 2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  Hence, in the sense of analytic continuation,  � ∞  −∞(1 − it)γ+p(1 + it)δ dt = 2sin πγ sin πδ  sin π(γ + δ)  � 1  −1(1 − t)γ+p(1 + t)δ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  The stated result now follows from the assumption that f (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xN) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) is a polynomial and so the evaluation of the multiple integrals reduces to the  one-dimensional integrals (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), which are related by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We can use Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 to extend Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 to the case that α, hence η, are complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let α be, in general, complex and recall that we choose η = −κ(N − 1) − 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For  κ = β/2 ∈ N,  (−1)κN(N−1)/2N (Cy)  N,β =  �  −2sin πα sin πα  sin π(α + α)  �N  N (sJ)  N,β  ���  a=b=−κ(N−1)−1−α,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  where the right-hand side is to be regarded as defined by its analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We observe that for β even, the product of differences |∆N(λ)|β in the definition of  the normalisation constants,  NN,β =  �  RN p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) dλ1 · · · dλN,  is a polynomial, and moreover,  (λk − λj)β = (−1)κ((1 − iλk) − (1 − iλj))β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The result now follows from the definition of the normalisation constants and the identity  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Analogous to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10), the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) can be rewritten as  SN(−κ(N − 1) − 1 − α, −κ(N − 1) − 1 − α, κ)  = (−1)κN(N−1)/2  �  −π sin π(α + α)  sin πα sin πα  �N N−1  ∏  j=0  Γ(2Re(α) + 1 + κj)Γ(1 + κ(j + 1))  Γ(α + 1 + κj)Γ(α + 1 + κj)Γ(1 + κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  79  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  This can be verified using the same steps as for equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10), and is known to be  true for β > 0 due yet again to Carlson’s theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case α is real, identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) reduces to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We can make use of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and a further application of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 in the  specification (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) of the eigenvalue densities ρ(Cy)(λ) and ρ(sJ)(λ) to deduce the analogue  of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 in the non-symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the setting of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2,  ρ(Cy)(iλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = − sin π(α + α)  2 sin πα sin πα ρ(sJ)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)  ���  a=b=−κ(N−1)−1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case α is real, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) reduces to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, there exist expressions in terms of orthogonal polynomials for  both sides of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) when β = 1, 2, or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The expressions for the right-hand  side can be found in [4], and the left-hand side in [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These expressions can be  checked to be consistent with equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), using the fact that the orthogonal poly-  nomials associated with the shifted Jacobi weight are the Jacobi polynomials P(a,b)  N  (λ),  while those associated with the Cauchy weight are the scaled Jacobi polynomials  i−NP(η,η)  N  (iλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Linear Differential Equations for the Eigenvalue Densities and  Resolvents  The Selberg integral theory in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 allows us to derive order β + 1 linear differential  equations satisfied by the eigenvalue densities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) of the Jacobi β ensembles with β = 2, 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  the β ↔ 4/β duality given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 then extends our results to β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Analogous to  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, these differential equations are homogeneous, and they have inhomogeneous  counterparts that are solved by the corresponding resolvents (recall that the resolvents have  multiple valid definitions, as introduced in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  After presenting the aforementioned differential equations for the Jacobi ensembles in  §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we use the theory of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to translate our results to the shifted Jacobi and  80  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  Cauchy ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We treat the β = 2 case in full generality and report on the β = 1, 4 cases  only in the symmetric case, since the non-symmetric β = 1, 4 differential equations have  cumbersome presentations with little expository benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The differential equations are all of  the same orders and simplify greatly in the symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They are given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, along  with brief discussions regarding connections to the circular Jacobi ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Extending the limiting procedures in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, in a similar fashion to what was seen for  the normalisation constants at the end of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, the Jacobi ensembles’ differential equations  transform into differential equations satisfied by the Laguerre ensembles’ eigenvalue densities  and resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We again treat β = 1, 2, and 4, and the differential equations are of the same  orders, but with simpler coefficients due to the loss of parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Repeating this exercise  to scale out both the parameters a and b results in the even simpler differential equations for  the Gaussian ensembles displayed in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rather than outlining this calculation,  we demonstrate an alternate extension of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that allows us to derive differential  equations for the Gaussian ensembles without needing to first make the Jacobi ensemble  analogues explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, we derive seventh order linear differential equations  that characterise the eigenvalue densities and resolvents of the β = 2/3 and β = 6 Gaussian  β ensembles, without any knowledge of the Jacobi β ensembles for these values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These  differential equations serve as a proof of concept: Our method applies for positive integer β  outside of the classical regime β = 1, 2, 4, and also allows us to access non-integer β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, the interest in the upcoming differential equations is that  comparable and/or equivalent (in the GOE, GSE, and classical unitary cases) results in the  works [221], [164], [151], [2], [319], [260], [23] have been shown therein and also in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  [223], [213] to be amenable to various forms of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The differential equations of this  section will be given two applications in this thesis: In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, we apply global and  edge scalings to the differential equations to give characterisations of the relevant ensembles  at these scaling limits, while in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we present linear recurrences for the spectral  moments mk of the classical matrix ensembles, supplementing the works [174], [224], [72],  among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another appeal of our derivations below is that they treat all of the classical  matrix ensembles through a unified method, in contrast to the aforementioned works whose  results we recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  81  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Differential equations for the Jacobi ensembles  Recall from §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 that the eigenvalue density ρ(J)(λ) is a scalar multiple of w(J)(λ) times  the function I(J)  N,β(λ) defined in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When β is an even integer, this latter  function is moreover related, via equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24), to the auxiliary function J(N)  β,0 (λ) defined  in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using the differential-difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) for the J(N)  β,p (λ), now  with p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , β, we derive a differential equation for J(N)  β,0 (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying straightforward  manipulations according to equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) then yields the sought differential  equations for ρ(J)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) with β ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The function J(N)  2,0 (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) satisfies the differential equation  0 = x2(x − 1)2 d3  dx3 J(N)  2,0 (x) − [3C2(x) − 2(1 − 2x)] x(x − 1) d2  dx2 J(N)  2,0 (x)  + [((a + N)(1 + 4N) + 4 + N) x(x − 1) + (2C2(x) − 3(1 − 2x)) C2(x)] d  dx J(N)  2,0 (x)  − 2N(a + N) [2C2(x) − 3(1 − 2x)] J(N)  2,0 (x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  where C2(x) = (a + 2N)(x − 1) − b − 2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Setting n = 2 and taking p = 0, 1, 2 in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17), we obtain the matrix differen-  tial equation  d  dx  �  ����  J(N)  2,0 (x)  J(N)  2,1 (x)  J(N)  2,2 (x)  �  ���� =  �  ����  A0x+B0  x(x−1)  2E0  x(x−1)  0  −D1  A1x+B1  x(x−1)  E1  x(x−1)  0  −D2  0  �  ����  �  ����  J(N)  2,0 (x)  J(N)  2,1 (x)  J(N)  2,2 (x)  �  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  The second row gives an expression for J(N)  2,2 (x) which transforms the third row into a  differential equation involving only J(N)  2,0 (x) and J(N)  2,1 (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This equation further transforms  into an equation for just J(N)  2,0 (x) upon substitution of the expression for J(N)  2,1 (x) drawn from  the first row:  0 = x2(x − 1)2 d3  dx3 J(N)  2,0 (x) −  � ˜C2,0(x) + ˜C2,1(x) + 1 − 2x  �  x(x − 1) d2  dx2 J(N)  2,0 (x)  +  �(D2E1 + 2D1E0 − A1 − 2A0 + 2)x(x − 1) + ˜C2,0(x) ˜C2,1(x)  � d  dx J(N)  2,0 (x)  +  �  A0 ˜C2,1(x) + (A1 − D2E1)(A0x + B0) − 2D1E0(1 − 2x)  �  J(N)  2,0 (x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  where ˜C2,p(x) = (Ap − 2)x + Bp + 1 with n = β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting the appropriate values for  the constants Ap, Bp, Dp and Ep gives the claimed result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  82  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now taking n = β = 4, the function J(N)  4,0 (x) satisfies the differential equation with  polynomial coefficients  0 = 4x4(x − 1)4 d5  dx5 J(N)  4,0 (x) − 20 [(a + 4N)(x − 1) − b − 2x] x3(x − 1)3 d4  dx4 J(N)  4,0 (x)  +  �  5(a + 4N)2 − 5a(a − 2) − 12  �  x3(x − 1)3 d3  dx3 J(N)  4,0 (x) + · · ·  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  where the (lengthier) specific forms of the coefficients of the lower order derivatives have been sup-  pressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (One may obtain the full expression by inverting the upcoming proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact,  this is the most efficient method of obtaining the full expression using computer algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like the preceding proof, setting n = 4 in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) and taking p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 4  yields the matrix differential equation  d  dx  �  ����������  J(N)  4,0 (x)  J(N)  4,1 (x)  J(N)  4,2 (x)  J(N)  4,3 (x)  J(N)  4,4 (x)  �  ����������  =  �  ����������  A0x+B0  x(x−1)  4E0  x(x−1)  0  0  0  −D1  A1x+B1  x(x−1)  3E1  x(x−1)  0  0  0  −D2  A2x+B2  x(x−1)  2E2  x(x−1)  0  0  0  −D3  A3x+B3  x(x−1)  E3  x(x−1)  0  0  0  −D4  0  �  ����������  �  ����������  J(N)  4,0 (x)  J(N)  4,1 (x)  J(N)  4,2 (x)  J(N)  4,3 (x)  J(N)  4,4 (x)  �  ����������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  For 1 ⩽ p ⩽ 4, the pth row gives an expression for J(N)  4,p (x) in terms of d  dx J(N)  4,p−1(x) and J(N)  4,k (x)  with k < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting these expressions (in the order of decreasing p) into the differential  equation corresponding to the fifth row yields a fifth order differential equation for J(N)  4,0 (x)  similar to that of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, we may easily obtain differential equations for I(J)  N,β(λ) for β = 2 and 4, which, in  turn, give us differential equations for ρ(J)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) for β = 1, 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define  D(J)  N,2 = x3(1 − x)3 d3  dx3 + 4(1 − 2x)x2(1 − x)2 d2  dx2  +  �(a + b + 2N)2 − 14  �  x2(1 − x)2 d  dx −  �  a2(1 − x) + b2x − 2  �  x(1 − x) d  dx  + 1  2  �(a + b + 2N)2 − 4  � (1 − 2x)x(1 − x) + 3  2  �  a2 − b2�  x(1 − x)  − a2(1 − x) + b2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  83  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  Then,  D(J)  N,2 ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  and  D(J)  N,2  1  NW(J)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) = (a + b + N)(a(1 − x) + bx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We change variables x �→ 1/x in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) to obtain a differential equation for  J(N)  2,0 (1/x) using the fact that  d  d(1/x) = −x2 d  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since this differential equation is homogeneous,  we can ignore constants of proportionality and substitute in  J(N)  2,0 (1/x) = x−a−2N(1 − x)−bρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, 2)  according to equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Repeatedly applying the product rule and then  replacing N + 1 by N gives equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying the Stieltjes transform to this equation  term by term (see Appendix B) and substituting in the values of the spectral moments m(J)  1  and m(J)  2  with β = 2 [242] yields equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) lowers the order by one relative to the fourth order differential  equation for ρ(J)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) given in the recent work [72, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With 1 ⩽ k ⩽ N, let ρ(w)  k  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) denote the k-point correlation function of  the random matrix ensemble corresponding to the classical weight w(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), as specified  by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is well known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 9]) that the generating function  E(w)  N ((s, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ) for the probabilities {E(w)  N,k(s, ∞)}N  k=0 of there being exactly k eigenvalues in  the interval (s, ∞) can be written in terms of the correlation functions according to  E(w)  N ((s, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ) = 1 +  N  ∑  k=1  (−ξ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  � ∞  s  dx1 · · ·  � ∞  s  dxk ρ(w)  k  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  The logarithmic derivative of these generating functions multiplied by simple polynomials  dependent on the weight w(λ) are known to be characterised by particular σ Painlev´e  equations [298], [136], [318], [137], [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It has been observed in [132, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3] that the third  order linear differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) for the eigenvalue densities of the  Gaussian and Laguerre unitary ensembles are equivalent to related σ Painlev´e equations  (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [119, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' An analogous result holds true for the differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  for ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, with v1 = v3 = N + (a + b)/2, v2 = (a + b)/2, v4 = (b − a)/2, in  studying the gap for the interval (0, s) one encounters the nonlinear equation [119, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='76)]  (t(1 − t) f ′′)2 − 4t(1 − t)( f ′)3 + 4(1 − 2t)( f ′)2 f + 4f ′ f 2 − 4f 2v2  1  +( f ′)2�  4tv2  1(1 − t) − (v2 − v4)2 − 4tv2v4  �  + 4f f ′(−v2  1 + 2tv2  1 + v2v4) = 0,  subject to the boundary condition f (t)  ∼  t→0+ −ξt(1 − t)ρ(J)(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting this bound-  ary condition for f and equating terms of order ξ2 shows that u(t) := t(1 − t)ρ(J)(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2)  satisfies the second order nonlinear differential equation  (t(1 − t)u′′(t))2 − 4(u(t))2v2  1 + (u′(t))2(4tv2  1(1 − t) − (v2 − v4)2 − 4tv2v4)  + 4u(t)u′(t)(−v2  1 + 2tv2  1 + v2v4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  Differentiating this and simplifying gives a third order linear differential equation that agrees  with equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling that κ := β/2, let  aβ :=  a  κ − 1,  bβ :=  b  κ − 1,  Nβ := (κ − 1)N  so that (a4, b4, N4) = (a, b, N) and (a1, b1, N1) = (−2a, −2b, −N/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 1 or 4, define  D(J)  N,β = 4x5(1 − x)5 d5  dx5 + 40(1 − 2x)x4(1 − x)4 d4  dx4 +  �  5˜c2 − 493  �  x4(1 − x)4 d3  dx3  −  �  5f+(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a, ˜b) − 88  �  x3(1 − x)3 d3  dx3 + 41  �˜a − ˜b  �  x3(1 − x)3 d2  dx2  +  �  19˜c2 − 539  � (1 − 2x)x3(1 − x)3 d2  dx2 − 22f−(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a, ˜b)x2(1 − x)2 d2  dx2  + 16(1 − 2x)x2(1 − x)2 d2  dx2 +  �  ˜c4 − 64˜c2 + 719  �  x3(1 − x)3 d  dx  −  �(˜c2 − 45)(˜a + ˜b − 6) + (˜a − ˜b)2 − 248  �  x2(1 − x)2 d  dx  −  �(˜c2 − 37)(˜a − ˜b)  � (1 − 2x)x2(1 − x)2 d  dx  +  �  f+(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a2, ˜b2) − 14f+(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a, ˜b) − 16  �  x(1 − x) d  dx  + 1  2  �  5(˜c2 − 9)(˜a − ˜b)  �  x2(1 − x)2 + 1  2(˜c2 − 9)2(1 − 2x)x2(1 − x)2  − 1  2  �(3˜c2 − 35) f−(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a, ˜b) + 7  2(˜a2 − ˜b2) + 4(˜a − ˜b)  �  x(1 − x)  − 1  2  �  4˜c2 − 36 + 3  2(˜a − ˜b)2� (1 − 2x)x(1 − x) + f−(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a2, ˜b2),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  85  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  where  ˜a = aβ(aβ − 2),  ˜b = bβ(bβ − 2),  ˜c = aβ + bβ + 4Nβ − 1,  f±(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜a, ˜b) = ˜a(1 − x) ± ˜bx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  Then, for β = 1 and 4,  D(J)  N,β ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  and  D(J)  N,β  1  NW(J)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = (˜c − 2Nβ)x(1 − x)  �  (aβ + bβ)(aβ + bβ − 2)(aβ(1 − x) + bβx)  + 4Nβ(˜c − 2Nβ)(2aβ(1 − x) + 2bβx − 1) − aβbβ(aβ + bβ − 6) − 4(aβ + bβ − 1)  �  − (˜c − 2Nβ)  �  aβ(aβ − 2)2(1 − x)2 + bβ(bβ − 2)2x2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The proof is done in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First, to see that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) holds for β = 4, one  undertakes the same steps as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' That is, change variables x �→ 1/x  in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), substitute in  J(N)  4,0 (1/x) = x−a−4N(1 − x)−bρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, 4),  and then replace N + 1 by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For the second step, apply the Stieltjes transform according  to Appendix B and substitute in the values of the spectral moments m(J)  1  to m(J)  4  [96], [242]  to obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) for β = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The third step proves equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) for β = 1 by employing  the β ↔ 4/β duality relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='98), which leads us to formulate equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) in terms  of the (aβ, bβ, Nβ) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since this step essentially redefines constants, applying the  inverse Stieltjes transform to this result returns equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) with the new constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the way that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) is presented, all of its dependencies on a, b and N are  captured by ˜a, ˜b and ˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, it can be seen that this operator is invariant under the  symmetry (x, a, b) ↔ (1 − x, b, a), which is a property of ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) have been checked for N = 1, 2 and to hold in the large N limit, while (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) have been checked to be consistent with expressions for the resolvent expansion  coefficients W(J),0  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , W(J),4  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) (recall Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) generated via the loop  equation analysis presented in [142] (see also §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 for a summary of the methods used in  86  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It should be noted that while the moments m(J)  1  to m(J)  4  of the Jacobi β ensemble’s  eigenvalue density are complicated rational functions of a, b, and N, the right-hand sides of  equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) are relatively simple polynomials, even though they are linear  combinations of these moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Differential equations for the shifted Jacobi and Cauchy ensembles  Upon making the simple change of variables x �→ (1 − x)/2 in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2,  analogous differential equations for the shifted Jacobi ensembles can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  outcomes are not different in any essential way, so no insight is gained from explicitly  presenting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Instead, we further enforce the restriction b = a so that significant  simplification occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the differential equations are for the eigenvalue densities and  resolvents of the symmetric shifted Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define  D(sJ)  N,2 = (1 − x2)3 d3  dx3 − 8x(1 − x2)2 d2  dx2  − 2(1 − x2)[3 − 2N2 − 4aN + (2(a + N)2 − 7)x2] d  dx  + 4x(a2 + 1 − N2 − 2aN + (a + N)2x2 − x2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  and for β = 1 and 4,  D(sJ)  N,β = 4(1 − x2)5 d5  dx5 − 80x(1 − x2)4 d4  dx4 + (5˜c2 − 493)(1 − x2)4 d3  dx3  − 4(5˜a − 88)(1 − x2)3 d3  dx3 + 16(11˜a − 8)x(1 − x2)2 d2  dx2 − 2(19˜c2 − 539)x(1 − x2)3 d2  dx2  + (˜c4 − 64˜c2 + 719)(1 − x2)3 d  dx − 8  �(˜c2 − 45)(˜a − 3) − 124  � (1 − x2)2 d  dx  + 16  �(˜a − 7)2 − 65  � (1 − x2) d  dx − (˜c2 − 9)2x(1 − x2)2  + 4  �  4(˜c2 − 9) + (3˜c2 − 35)˜a  �  x(1 − x2) − 32˜a2x,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  taking ˜a, ˜c as defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, with b = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for β = 1, 2, and 4, we have  D(sJ)  N,β ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)|a=b = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  87  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  and  D(sJ)  N,β  1  NW(sJ)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)|a=b  =  �  �  �  �  �  �  �  �  �  �  �  �  �  4a(2a + N),  β = 2,  8(2Nβ + 2aβ − 1)  �  2a3  βx2 − 2(2Nβ + 1)(Nβ − 1)(x2 − 1)  +a2  β((8Nβ − 3)x2 − 8Nβ − 5) + 8aβ(Nβ(Nβ − 1)(x2 − 1) + 1)  �  ,  β = 1, 4,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  where aβ, Nβ are also as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) with β = 2 is a generalisation of that given in [151, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4] for  the β = 2 Legendre ensemble, which is the sJUE with a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  and calculations similar to those seen in Appendix B, the differential equations satisfied by  ρ(Cy)(x) and W(Cy)  1  (x) for β even and α > −1/2 real can be obtained from those satisfied by  ρ(sJ)(x)|a=b and W(sJ)  1  (x)|a=b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is done by setting a = −κ(N − 1) − 1 − α and  replacing x by ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Doing so in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 gives third and fifth order linear differential  equations for β = 2 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 tells us that the latter  differential equations are valid for β = 1 after reparametrising properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let α > −1/2 be real so that we are restricted to the symmetric Cauchy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Recalling the definition of Nβ in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, let  αβ :=  α  κ − 1,  ˜α := αβ(αβ − 1),  ˜N := (2Nβ + αβ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Define  D(Cy)  N,2 = (1 + x2)3 d3  dx3 + 8x(1 + x2)2 d2  dx2 + 2(1 + x2)[3 + 2N(N + 2α) + (7 − 2α2)x2] d  dx  + 4x(1 + α2 + 2N(N + 2α) + (1 − α2)x2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  and for β = 1 and 4,  D(Cy)  N,β = 4(1 + x2)5 d5  dx5 + 80x(1 + x2)4 d4  dx4 − 4(5˜α − 122)(1 + x2)4 d3  dx3  + 4(5 ˜N − 93)(1 + x2)3 d3  dx3 − 8(19˜α − 130)x(1 + x2)3 d2  d2x + 16(11 ˜N − 19)x(1 + x2)2 d2  d2x  + 8(2˜α2 − 31˜α + 82)(1 + x2)3 d  dx − 32  �(˜α − 11)( ˜N − 4) − 31  � (1 + x2)2 d  dx  + 16  �( ˜N − 8)2 − 65  � (1 + x2) d  dx + 16(˜α − 2)2x(1 + x2)2  − 16  �(3˜α − 8) ˜N + ˜α  �  x(1 + x2) + 32( ˜N − 1)2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  88  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  Then, for β = 1, 2, and 4, with α > −1/2 real, we have  D(Cy)  β,N ρ(Cy)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  and  D(Cy)  N,β  1  NW(Cy)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)  =  �  �  �  �  �  �  �  �  �  �  �  �  �  4(N + α)(N + 2α),  β = 2,  8(2Nβ + 2αβ − 1)  �  (2Nβ + 1)(8N2  β + x2 − 1) − 2αβ(1 + x2α2  β)  +4Nβαβ(6Nβ + x2 + 3) + α2  β(3x2 − 4Nβ(x2 − 2) + 5)  �  ,  β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) with β = 2 was recently derived in [23] using a method of Ledoux [223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 the relationship between the generating function  E(w)  N ((s, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ) and k-point correlation functions ρ(w)  k  recounted therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We now supplement  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 with its counterpart for the Cauchy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case β = 2, it is known [318] that  σ(s) := (1 + s2) d  ds log E(Cy)  N  ((s, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  satisfies the nonlinear equation (which can be identified in terms of the σ-PVI equation [138];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  see also equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) below)  (1 + s2)2(σ′′)2 + 4(1 + s2)(σ′)3 − 8sσ(σ′)2  + 4σ2(σ′ − α2) + 8α2sσσ′ + 4[N(N + 2α) − α2s2](σ′)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  Note that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) is independent of the parameter ξ in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  According to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), to leading order in ξ,  σ(s) = ξr(s) + O(ξ2),  r(s) := (1 + s2)ρ(Cy)  1  (s) = (1 + s2)ρ(Cy)(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  Substituting in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) and equating terms to the leading order in ξ (which is O(ξ2)) shows  (1 + s2)2(r′′(s))2 − 4α2(r(s))2 + 8α2sr(s)r′(s) + 4[N(N + 2α) − α2s2](r′(s))2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  Upon differentiating with respect to s, a factor of r′′(s) can be cancelled and a third order  linear differential equation results,  (1 + s2)2r′′′(s) + 2s(1 + s2)r′′(s) + 4[N(N + 2α) − α2s2]r′(s) + 4α2sr(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  89  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  Recalling the definition of r(s) in terms of ρ(Cy)(s) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), we see that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) is  equivalent to the third order differential equation given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The β = 1, 4 analogue of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27), obtained by taking ρ(Cy)(s) = r(s)/(1 + s2) in  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), is  (1 + s2)4r(5)(s) + 10s(1 + s2)3r(4)(s) − (5˜α − 22)(1 + s2)3r′′′(s)  + (5 ˜N − 13)(1 + s2)2r′′′(s) − 8(˜α − 1)s(1 + s2)2r′′(s)  + 2(7 ˜N + 1)s(1 + s2)r′′(s) + 4˜α2(1 + s2)2r′(s) − 2  �(4˜α − 1) ˜N + 1  � (1 + s2)r′(s)  + 4( ˜N − 1)2r′(s) − 4˜α2s(1 + s2)r(s) + 4˜α( ˜N − 1)sr(s) = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  where ˜α and ˜N are as given in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In comparing equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  to their counterparts in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7, we see that degree of each of the coefficients (which  alternate between being even and odd in s) has been reduced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that the the Cauchy and circular Jacobi ensembles are related  by the Cayley transformation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) and stereographic projection λi = cot(θi/2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In  particular, the eigenvalue densities of the two ensembles are related by  ρ(Cy)(λ) = 2 sin2(θ/2) ρ(cJ)(eiθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  Equivalently, in the notation r(s) of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) with s = cot(θ/2),  r(s) = 2ρ(cJ)(eiθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  In the case α = 0, the circular Jacobi weight w(cJ)(eiθ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) is a constant and so according  to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30), r(s) is then also a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We can see immediately that this is consistent  with equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Suppose now θ is scaled by writing θ = 2X/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, in the scaling limit N → ∞,  the circular Jacobi ensemble exhibits a spectrum singularity at θ = 0, in keeping with the  discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In view of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30), in the cases β = 1, 2, and 4, differential  equations for the corresponding density ρ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')(X) can be obtained by setting s = N/X and  r(s) = ρ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')(X) in equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28), and then equating terms at leading order  in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Specifically for β = 2, we therefore have that R(X) = ρ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')(X) satisfies the third order  linear differential equation  X2R′′′(X) + 4XR′′(X) + (2 − 4α2 + 4X2)R′(X) − 4α2  X R(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  90  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  This is consistent with the known exact formula [257], [119, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) with πρ = 1]  ρ(s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')(x) = x  2  �  (Jα−1/2(x))2 + (Jα+1/2(x))2 − 2α  x Jα−1/2(x)Jα+1/2(x)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  as can be checked using computer algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Though there is no present benefit in presenting the analogue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 for the  non-symmetric shifted Jacobi ensemble, there is merit in deriving the differential equation  satisfied by the eigenvalue density of the non-symmetric CyUE, especially considering its  relatively compact form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Repeating the process outlined above, we first make the change  of variables x �→ (1 − x)/2 in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) to obtain a differential equation for the  non-symmetric shifted Jacobi unitary ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Setting a = −N − α, b = −N − α, and  replacing x by ix in this differential equation, it follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 that we obtain  the differential equation satisfied by ρ(Cy)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define the third order differential operator  ˜D(Cy)  N,2 = (1 + x2)3 d3  dx3 + 8x(1 + x2)2 d2  dx2  + (1 + x2)[6 + 4N(N + α + α) + (α − α)2 + 2i(α − α)(2N + α + α)x + (14 − (α + α)2)x2] d  dx  + [4 + 8N(N + α + α) + 3α2 − 2αα + 3¯α2 + (4 − (α + α)2)x2]x  + i(α − α)(2N + α + α)(3x2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  For β = 2 and α ∈ C with constraint Re(α) > −1/2, we have  ˜D(Cy)  N,2 ρ(Cy)(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' After a simple change of variables, the Jimbo–Miwa–Okamoto σ-form of the  Painlev´e differential equation reads [138, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)]  h′�  (1 + t2)h′′�2  + 4  �  h′(h − th′) − ib1b2b3b4  �2  + 4  4  ∏  k=1  (h′ + b2  k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  Let b = (b1, b2, b3, b4) and define e′  2[b], e2[b] as the elementary symmetric polynomials of  degree two in {b1, b3, b4} and {b1, b2, b3, b4}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Set  U(Cy)  N  (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (α1, α2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ) = (t2 + 1) d  dt log  �  (it − 1)e′  2[b]−e2[b]/2(it + 1)e2[b]/2E(Cy)  N  ((t, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ))  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  91  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  where E(Cy)  N  is specified by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) in the non-symmetric case with α = α1 + iα2  (α2 ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have from [138, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 15] that U(Cy)  N  satisfies the transformed σ-PVI equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) with parameters  b = (−α1, −iα2, N + α1, α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  Analogous to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), we have from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) that  U(Cy)  N  (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (α1, α2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ξ) = (N + α1)α2 − α2  1t + ξr(t) + O(ξ2),  r(t) := (1 + t2)ρ(Cy)(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) and equating terms of leading order in ξ, which  as for equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) occurs at order ξ2, then differentiating and cancelling a factor of r′′  shows  (1 + t2)2r′′′ + 2t(1 + t2)r′′ + 4[((N + α1)α2 + α2  1t)(r − tr′)  + ((N + α1)2 − (N + α1)α2t − α2  1 − α2  2)r′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  In the case α2 = 0, this agrees with equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now, substituting for r(t) in terms of  ρ(Cy)(t) as specified in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) reclaims equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Differential equations for the Laguerre ensembles  As mentioned at the beginning of this section, the eigenvalue density ρ(L)(x) of the Laguerre  β ensemble can be obtained from that of the Jacobi β ensemble via a limiting procedure:  From equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), upon extending the idea of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we see (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12))  ρ(L)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β) = lim  b→∞ b(N+3)Nκ+(N+2)a+N+1 ρ(J)� x  b ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  This fact allows one to transform differential equations satisfied by ρ(J)(x) into analogues  satisfied by ρ(L)(x), which will be presented in a moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As an aside, suppose that one would like to obtain differential equations for ρ(L)(x)  without prior knowledge of the analogous differential equations for ρ(J)(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', if the results  of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 were not available, or if one were interested in ensembles with β /∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, it  is actually more efficient to circumvent computation of the differential equations for ρ(J)(x)  and use the aforementioned limiting procedure indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' That is, let  L(N)  β,p (x) = lim  b→∞  �− x  b  �βN+p J(N)  β,p (b/x),  ˜L(N)  β,p (x) = xae−x L(N)  β,p (x)  92  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  so that by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), ρ(L)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β) is proportional to ˜L(N)  β,0 (x) when β is a positive  even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) gives a differential-difference equation for  �− x  b  �βN+p J(N)  β,p (b/x)  and taking the limit b → ∞ thus gives a differential-difference equation for L(N)  β,p (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Sub-  stituting L(N)  β,p (x) = x−aex ˜L(N)  β,p (x) then gives a differential-difference equation for ˜L(N)  β,p (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  These equations with p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , β are equivalent to matrix differential equations not unlike  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, having taken the limit b → ∞, we ensure that these new matrix  differential equations are simpler, and have the added benefit of simplifying down to scalar  differential equations for ρ(L)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β) rather than for auxiliary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One may then  apply the Stieltjes transform to obtain differential equations for the resolvents, and use the  β ↔ 4/β dualities given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 to obtain mirror differential equations like those seen  in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since we are interested in ρ(L)(x) with β ∈ {1, 2, 4} and we have differential equations for  ρ(J)(x) for these β values, we use the more direct approach to give the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Retaining the definitions of aβ, Nβ, and ˜a from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, define  D(L)  2,N = x3 d3  dx3 + 4x2 d2  dx2 −  �  x2 − 2(a + 2N)x + a2 − 2  �  x d  dx +  �(a + 2N)x − a2�  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  and for β = 1 or 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  D(L)  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='β = 4x5 d5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='dx5 + 40x4 d4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='dx4 −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='− 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1 + 5˜a − 88  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x3 d3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='dx3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='− 38  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1 + 22˜a − 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x2 d2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='dx2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='− 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='+ 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�2 + ˜a − 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='(κ − 1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4(˜a − 3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1 − ˜a2 + 14˜a + 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='dx −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�2 + ˜a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='− (3˜a + 4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='aβ + 4Nβ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='κ − 1 + ˜a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  Then, for β = 1, 2, and 4,  D(L)  N,β ρ(L)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43)  93  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  and  D(L)  N,β  1  NW(L)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (x + a),  β = 2,  4  κ−1  �  2  �  x  κ−1  �2 + (2aβ − 1)  x  κ−1  �  Nβ  −  1  κ−1  ��  x  κ−1  �3 − (aβ + 2)  �  x  κ−1  �2�  +  1  κ−1  �  (a2  β + 4aβ − 4)  x  κ−1 − aβ(aβ − 2)2�  ,  β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13), change variables x �→ x/b, mul-  tiply both sides by b(N+2)(N−1)κ+(N+1)a+N according to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40), and then take the  limit b → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is equivalent to changing variables x �→ x/b, extracting terms of leading  order in b, then rewriting ρ(J) as ρ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To obtain equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44), apply the same prescription to equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Alternatively, one may apply the Stieltjes transform to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is considerably  easier than in the Jacobi case (see Appendix B) since  � ∞  0  xn  s − x  dn  dxn ρ(L)(x) dx = sn dn  dsn W(L)  1  (s)  can be computed through repeated integration by parts using the fact that the boundary  terms vanish at all stages: For a > −1 real and m > n non-negative integers,  xm  s−x  dn  dxn ρ(L)(x)  has a factor of xa+m−n which dominates at x = 0 and a factor of e−x which dominates as  x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) with β = 2 is equivalent to that seen in [164], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When β = 1, 4, it has  been checked for N = 1, 2 using computer algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From inspection, it seems that the natural  variable of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) is x/(κ − 1), which is the limit limb→∞ bβ  � x  b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  is in keeping with the duality relation on the resolvent presented in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Strictly  speaking, changing variables to y = x/(κ − 1) is not natural and would be counterproductive  since the corresponding weight [(κ − 1)y]a e(1−κ)y vanishes at κ = 1 and has different support  depending on whether κ < 1 or κ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  Differential equations for the Gaussian ensembles  The previous subsection contains a discussion on how one would obtain differential equations  for the densities and resolvents of Laguerre ensembles with even β without knowledge of  94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  differential equations for the corresponding Jacobi β ensembles’ densities and resolvents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We now elucidate those ideas by explicitly applying them to the study of the Gaussian  ensembles with β = 6 and consequently, by duality, β = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, since we do not have  differential equations for the β = 6 Jacobi ensemble at hand, we cannot immediately apply  the direct limiting approach used in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, for our purposes, it  is in fact more efficient to replicate the proofs of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 while taking limits at an earlier stage  so as to circumvent the need for investigating the β = 6 Jacobi ensemble altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Like in the Jacobi and Laguerre cases, our initial focus is the average  I(G)  N,β(λ) :=  �  N  ∏  i=1  |λ − λi|β  �  GEN,β  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45)  where the subscript GEN,β indicates that the average is with respect to the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) with weight w(G)(λ) = e−λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Gaussian analogue of the duality relation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) is [27]  �  N  ∏  i=1  �  λ − κ−1/2λi  �n  �  GEN,β  =  �  n  ∏  i=1  (λ − iλi)N  �  GEn,4/β  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  Replacing λ by κ−1/2λ in the above duality relation and then factoring (−i)nN from the  right-hand side shows that for even β, I(G)  N,β(λ) is proportional to G(N)  β,0 (λ), where  G(N)  n,p (λ) := (−i)nN+p  �  n  ∏  i=1  �  λi + iκ−1/2λ  �N+χi⩽p  �  GEn,4/β  ,  0 ⩽ p ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  Setting a′ = b′ = L and changing variables λi �→ 1  2  �  1 + λi  √  L  �  in J(N)  n,p (λ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), we see that  G(N)  n,p (λ) = lim  L→∞ 4nL(−2i  √  L)nN+p(2  √  L)n(n−1)/κ+n J(N)  n,p  �  1  2  �  1 − i  �  2  βLλ  �����  a′=b′=L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48)  Thus, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) simplifies to a differential-difference equation for G(N)  n,p (λ),  (n − p)G(N)  n,p+1(λ) = (n−p)  √κ λG(N)  n,p (λ) −  √κ  2  d  dλG(N)  n,p (x)  + p  2  � 1  κ(n − p) + N + 1  �  G(N)  n,p−1(λ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49)  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [133, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking the L → ∞ limit early has already yielded a simpler equation  than (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we can take this one step further by defining  ˜G(N)  n,p (λ) = e−λ2G(N)  n,p (λ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50)  95  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  so that ρ(G)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β) is proportional to ˜G(N)  β,0 (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is then easy to obtain the differential-  difference equation  d  dλ  ˜G(N)  n,p (λ) =  p  √κ  � 1  κ(n − p) + N + 1  � ˜G(N)  n,p−1(λ)  + 2  � 1  κ(n − p) − 1  �  λ ˜G(N)  n,p (λ) −  2  √κ(n − p) ˜G(N)  n,p+1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)  With n = β ∈ 2N and p = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n, this is equivalent to a matrix differential equation on  the vector v =  �  ˜G(N)  n,0 (x) · · · ˜G(N)  n,n (x)  �T  , which is moreover equivalent to a scalar differential  equation for ρ(G)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2/3 or 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' define  D(G)  N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='β = 81(κ − 1)7/2 d7  dx7 + 1008  �  3Nβ − 2x2  κ − 1 + 2  �  (κ − 1)5/2 d5  dx5  + 2016x(κ − 1)3/2 d4  dx4  + 64  �  21Nβ − 14x2  κ − 1 + 5  � �  21Nβ − 14x2  κ − 1 + 23  �  (κ − 1)3/2 d3  dx3  + 9984  �  3Nβ − 2x2  κ − 1 + 2  �  x(κ − 1)1/2 d2  dx2  + 256  �  54Nβ  �  4N2  β + 8Nβ + 3  �  − (432N2  β + 576Nβ + 57) x2  κ − 1  + 96(3Nβ + 2)  x4  (κ − 1)2 −  64x6  (κ − 1)3 − 20  �  (κ − 1)1/2 d  dx  + 256  �  144N2  β + 192Nβ − 64(3Nβ + 2) x2  κ − 1 +  64x4  (κ − 1)2 + 25  �  x  (κ − 1)1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  where we retain the definition Nβ = (κ − 1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for these same β values,  D(G)  N,β ρ(G)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) = 0  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53)  and  D(G)  N,β  1  NW(G)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) =  211  √  κ − 1  � 4x2  κ − 1 − 6Nβ − 7  �2  − 3  212  √  κ − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Take equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) with n = β = 6 and N replaced by N − 1 to obtain the matrix  96  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Linear Differential Equations for the Eigenvalue Densities and Resolvents  differential equation  dv  dx =  �  �����������������  2x  −4  √  3  0  0  0  0  0  3N+5  3  √  3  4x  3  − 10  √  3  0  0  0  0  0  6N+8  3  √  3  2x  3  − 8  √  3  0  0  0  0  0  √  3(N + 1)  0  −2  √  3  0  0  0  0  0  12N+8  3  √  3  − 2x  3  − 4  √  3  0  0  0  0  0  15N+5  3  √  3  − 4x  3  − 2  √  3  0  0  0  0  0  2  √  3N  −2x  �  �����������������  v  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='55)  for v =  �  ˜G(N−1)  6,0  (x) · · · ˜G(N−1)  6,6  (x)  �T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like in the proofs of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, for 1 ⩽ p ⩽ 6,  the pth row of the above matrix differential equation gives an expression for ˜G(N−1)  6,p  (x) in  terms of  d  dx ˜G(N−1)  6,p−1 (x) and ˜G(N−1)  6,k  (x) with k < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting these expressions into the  equation corresponding to the last row in the order of decreasing p then yields a seventh-  order differential equation satisfied by ˜G(N−1)  6,0  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since ρ(G)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 6) is proportional to  ˜G(N−1)  6,0  (x), this equation is equivalent to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) for β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking the Stieltjes transform of  this result and substituting in the spectral moments m(G)  2  and m(G)  4  from [319] then yields  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54) for β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Employing the duality relation given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 for the  resolvent then shows that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54) also holds for β = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, taking the inverse  Stieltjes transform of this result shows that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) holds for β = 2/3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) has been checked for N = 1, 2 using computer algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar to  the Laguerre case, it seems like x/√  κ − 1 is the natural variable in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  is evidently due to the duality relation used in the proof of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like in the  Laguerre case, there is presently no benefit in changing variables to x/√  κ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It has been mentioned that equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) leads to a matrix differential equation which  is equivalent to a scalar differential equation for ρ(G)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N + 1, β) when β is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2  and 4, these differential equations are respectively  d  dx  �  ����  ˜G(N−1)  2,0  (x)  ˜G(N−1)  2,1  (x)  ˜G(N−1)  2,2  (x)  �  ���� =  �  ����  2x  −4  0  N + 1  0  −2  0  2N  −2x  �  ����  �  ����  ˜G(N−1)  2,0  (x)  ˜G(N−1)  2,1  (x)  ˜G(N−1)  2,2  (x)  �  ����  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56)  97  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  and  d  dx  �  ����������  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  (x)  �  ����������  =  �  ����������  2x  −4  √  2  0  0  0  2N+3  2  √  2  x  −3  √  2  0  0  0  √  2(N + 1)  0  −2  √  2  0  0  0  6N+3  2  √  2  −x  −  √  2  0  0  0  2  √  2N  −2x  �  ����������  �  ����������  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  (x)  ˜G(N−1)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  (x)  �  ����������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57)  The corresponding scalar differential equations for the respective eigenvalue densities agree  with those of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 after scaling x �→  �  Nκ  2g x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' So too does the differential equation  for ρ(G)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 1) obtained through the β ↔ 4/β duality given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  Scalings of the Differential Equations  The differential equations of the previous section hold true not only for finite integers N,  but also in the large N limit, upon appropriate scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this section we consider two  scaling regimes: In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we study the classical matrix ensembles (and also the Gaussian  β ensembles with β = 2/3, 6) under the global scalings introduced in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6, while  in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we look at the soft and hard edge scaling regimes (recall the discussion following  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the global scaling regime, we use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to expand the scaled resolvents ˜W1(x),  specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24), in 1/N and derive first order differential equations which  characterise the expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Being first order, these differential equations are  straightforward to solve, and using the Sokhotski–Plemelj inversion formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), we  are able to compute expressions for the large N limiting forms ρ0(λ) of the smoothed  eigenvalue densities and their 1/N corrections ρ1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As expected, we recover the Wigner  semi-circle, Marˇcenko–Pastur, and Wachter laws seen in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, along with the  expression for ρ(Cy),0(λ) presented therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, in keeping with Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 following  this proposition, we observe that while the large N limiting forms ρ0(λ) are β-independent,  the 1/N corrections ρ1(λ) are different for each β that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is in keeping with  the general-β results known from loop equation analysis (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [319], [142] and references  therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  98  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  In contrast to the β-universality seen for the large N limiting forms of the global scaled  eigenvalue densities, we see a different type of universality at the soft and hard edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Namely, it is known [115], [256] that for fixed β, a soft edge of a classical β ensemble behaves  statistically as that of any other classical β ensemble, in the large N limit — likewise for a  hard edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we may denote the large N limiting forms of the soft and hard edge scaled  eigenvalue densities of the classical β ensembles as ρ(so f t)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) and ρ(hard)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we apply these scalings to the differential equations of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 to derive linear  differential equations satisfied by ρ(so f t)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) for β ∈ {2/3, 1, 2, 4, 6} and ρ(hard)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) for  β ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Global scaled differential equations  Global scaling of the eigenvalue density refers to a choice of scaling which ensures that, in  the large N limit, the scaled density is supported on a finite interval on which it integrates to  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Such a transformation is not unique, due to the length of the interval of support being  unspecified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for convenience, we follow the conventions outlined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6, which  are chosen to be consistent with the earlier works [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We recall then that the global  scaled eigenvalue densities of the classical β ensembles are  ˜ρ(G)(λ) =  �  κ  N ρ(G)(  √  κNλ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  ˜ρ(L)(λ) = κ ρ(L)(κNλ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  ˜ρ(J)(λ) = 1  N ρ(J)(λ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  ˜ρ(Cy)(λ) = 1  N ρ(Cy)(λ)  ���  α=ˆακN,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  with corresponding scaled resolvents (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24),  ˜W(G)  1  (x) :=  √  κN W(G)  1  (  √  κNx),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  ˜W(L)  1  (x) := κN W(L)  1  (κNx),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  ˜W(Cy)  1  (x) := W(Cy)  1  (x)  ���  α=ˆακN,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  where ˆα is constant in N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' since the spectrum of the Jacobi β ensemble is already supported  on [0, 1], there is no need to scale its resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, it was mentioned that the scaled  resolvents are more robust than the global scaled eigenvalue densities in the sense that they  99  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  admit well-defined large N expansions due to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, following [319], [142]  again for consistency, we consider the expansions  ˜W(G)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) = 2N  ∞  ∑  l=0  W(G),l  1  (x)  (2√κN)l ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  ˜W(L)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) = N  ∞  ∑  l=0  W(L),l  1  (x)  (√κN)l ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  W(J)  1 (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) = N  ∞  ∑  l=0  W(J),l  1  (x)  (κN)l  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  ˜W(Cy)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) = N  ∞  ∑  l=0  W(Cy),l  1  (x)  (κN)l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  Substituting these expansions into the differential equations of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 (after scaling  according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)) and equating terms of like order in N results in linear differential  equations for the expansion coefficients Wl  1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The differential equations for Wl  1(x) are  first order with inhomogeneous terms dependent on {Wk  1(x)}l−1  k=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, they constitute  a recursive process for computing the scaled resolvents ˜W1(x) up to any desired order in  1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The topological recursion or, equivalently, loop equation formalism [110] (see also the  review given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) accomplishes the same task for general β > 0, but our approach,  valid for the specific β values considered, is more efficient since there is no need to consider  the n-point correlators Un, Wn for n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As a consequence, our work is able to isolate extra  structures, such as those studied by Haagerup and Thorbjørnsen [170]: In [170], the authors  showed in the GUE case that W(G),l  1  (x) is a polynomial in (x2 − 2)−1/2, with the polynomial  coefficients satisfying a certain three-term recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Differential equations for W(G),l  1  (x)  We begin with the Gaussian β ensembles with β ∈ {2/3, 1, 4, 6} and refer to [170] for the  β = 2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We reuse the notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 with g = 1/2 so that h = √κ − 1/√κ  and y(G) :=  √  x2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for β = 1 and 4, the expansion coefficients of ˜W(G)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  satisfy the differential equations  y2  (G)  d  dxW(G),0  1  (x) − xW(G),0  1  (x) = −1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  100  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  y2  (G)  d  dxW(G),1  1  (x) − xW(G),1  1  (x) = 4h d  dxW(G),0  1  (x) −  h  y2  (G)  �  2xW(G),0  1  (x) + 5  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  and for l ⩾ 2, the general differential equation  y2  (G)  d  dxW(G),l  1  (x) − xW(G),l  1  (x) = 4h d  dxW(G),l−1  1  (x) − 2hx  y2  (G)  W(G),l−1  1  (x)  +  1  y2  (G)  �5y2  (G)  2  d3  dx3 − 3x d2  dx2 + d  dx  �  W(G),l−2  1  (x)  − 5h  y2  (G)  d3  dx3W(G),l−3  1  (x) +  1  y2  (G)  d5  dx5W(G),l−4  1  (x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  where we set W(G),k  1  := 0 for k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Beginning with the β = 1, 4 case of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), set g = 1/2 and map x �→  √  κNx  to obtain a differential equation for ˜W(G)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substitute in the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and  equate terms of like order in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The terms of leading order in N correspond to equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12), the next to leading order terms correspond to equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us recall from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 the relationship between the resolvent expansion coefficients  {Wl  1(x)}∞  l=0 and the coefficients { ˜Mk,l}∞  l=0 of the 1/N expansions of the spectral moments ˜mk  of ˜ρ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The facts that ˜m0 = m0/N = 1 and Wl  1(x) has asymptotic expansion ∑∞  k=0 ˜Mk,l/xk+1  tell us that the solutions of the differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) are fully determined by  the boundary conditions at large x,  W0  1(x) = 1  x + O  � 1  x2  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  Wl  1(x) = O  � 1  x2  �  ,  l > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  This is in fact true for all of the first order differential equations given in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, applying the above proof to differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54) gives analogous differential  equations for the expansion coefficients W(G),l  1  (x) when β = 2/3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Retain the definition of y(G) from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for β = 2/3 and 6, the  expansion coefficients of ˜W(G)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) satisfy the differential equations  y2  (G)  d  dxW(G),0  1  (x) − xW(G),0  1  (x) = −1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  101  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  y2  (G)  d  dxW(G),1  1  (x) − xW(G),1  1  (x) = 6h d  dxW(G),0  1  (x) −  h  y2  (G)  �  4xW(G),0  1  (x) − 7  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  y2  (G)  d  dxW(G),2  1  (x) − xW(G),2  1  (x) = 6h d  dxW(G),1  1  (x) − 4hx  y2  (G)  W(G),1  1  (x) −  43  3y4  (G)  +  1  12y4  (G)  �  49y4  (G)  d3  dx3 − 78xy2  (G)  d2  dx2 + 3(72 − 19x2) d  dx + 25x  �  W(G),0  1  (x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  and for l ⩾ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the general differential equation  y2  (G)  d  dxW(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l  1  (x) − xW(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l  1  (x) = 6h d  dxW(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−1  1  (x) − 4hx  y2  (G)  W(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−1  1  (x)  +  1  12y4  (G)  �  49y4  (G)  d3  dx3 − 78xy2  (G)  d2  dx2 + 3(72 − 19x2) d  dx + 25x  �  W(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  +  h  3y4  (G)  �  49y2  (G)  d3  dx3 + 39x d2  dx2 − 10 d  dx  �  W(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−3  1  + 7h  y4  (G)  d5  dx5W(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−5  1  +  1  18y4  (G)  �  63y2  (G)  d5  dx5 + 63x d4  dx4 + 230 d3  dx3  �  W(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−4  1  +  3h  4y4  (G)  d7  dx7W(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−6  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  where we again set W(G),k  1  := 0 for k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  This proposition has been checked against [319] up to l = 6, and thus also serves as a  check for differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54) up to order six in 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar checks for consistency,  against [142], have been carried out for Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Differential equations for W(L),l  1  (x) and W(J),l  1  (x)  Since the differential equations for the Laguerre and Jacobi ensembles’ resolvents involve the  parameters a, b, equating terms of equal order in N requires us to choose how these parame-  ters depend on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To make our results suitable for most known or potential applications  [33], [56], [305], [266], [229], [240], [241], [70], [71] and to improve clarity, we take a = ˆaκN  and b = ˆbκN with ˆa, ˆb constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that our method easily accommodates for more  general N-expansions of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Set a = ˆaκN with ˆa constant in N, and let y(L) :=  �  (ˆa − x)2 − 4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  expansion coefficients of the LUE scaled resolvent ˜W(L)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) satisfy the differential equation  − xy2  (L)  d  dxW(L),0  1  (x) +  �(ˆa + 2)x − ˆa2�  W(L),0  1  (x) = x + ˆa,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  102  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  and for l ⩾ 2, the general differential equation  xy2  (L)  d  dxW(L),l  1  (x) −  �(ˆa + 2)x − ˆa2�  W(L),l  1  (x)  =  �  x3 d3  dx3 + 4x2 d2  dx2 + 2x d  dx  �  W(L),l−2  1  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Consider equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) with β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In it, set a = ˆaN and map x �→ κNx to obtain a  differential equation for ˜W(L)  1  (x)  ��  a=ˆaN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substitute in the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and equate terms  of equal order in N to extract differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the κ = β/2 = 1, a = ˆaN setting considered here, the operator D(L)  N,2  ��  x�→κNx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  is even in N while the right-hand side of differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) under the mapping  x �→ κNx is odd in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It follows that ˜W(L)  1  (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) is an odd function of N, so W(L),k  1  (x) = 0  when k is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, differential-difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) holds vacuously for odd l, and  should otherwise be interpreted as a recursion over even l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note also that the vanishing  of W(L),k  1  (x) for k odd implies that the spectral moments ˜m(L)  k  (recall §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) of the scaled  eigenvalue density ˜ρ(L)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) are hence even functions of N, in keeping with the second  point of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the β = 1, 4 analogue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13 given below, which is  obtained by repeating the above proof starting with the β = 1, 4 cases of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  and taking a = ˆaκN, such phenomena are not seen since W(L),k  1  (x) is non-zero for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Retain the definitions of h, ˆa and y(L) from Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 1  and 4, the expansion coefficients of the scaled resolvent ˜W(L)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) satisfy the differential equations  − xy2  (L)  d  dxW(L),0  1  (x) +  �(ˆa + 2)x − ˆa2�  W(L),0  1  (x) = x + ˆa,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  xy4  (L)  d  dxW(L),1  1  (x) − y2  (L)  �(ˆa + 2)x − ˆa2�  W(L),1  1  (x)  = 4hˆaxy2  (L)  d  dxW(L),0  1  (x) + 2hˆa  �  x2 − 3(ˆa + 2)x + 2ˆa2�  W(L),0  1  (x)  + 2h  �  x(x − 1) + 2ˆax + 2ˆa2�  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  103  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  xy4  (L)  d  dxW(L),2  1  (x) − y2  (L)  �(ˆa + 2)x − ˆa2�  W(L),2  1  (x)  = 4hˆaxy2  (L)  d  dxW(L),1  1  (x) + 2hˆa  �  x2 − 3(ˆa + 2)x + 2ˆa2�  W(L),1  1  (x)  + 5  2x3y2  (L)  d3  dx3W(L),0  1  (x) + x2 �  8x2 − 19(ˆa + 2)x + 11ˆa2� d2  dx2W(L),0  1  (x)  + x  �  2x2 − 6(ˆa + 2)x + 5ˆa2� d  dxW(L),0  1  (x)  + 2  �(ˆa + 2)x − ˆa2�  W(L),0  1  (x) − 2(ˆa + x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  and for l ⩾ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the general differential equation  xy4  (L)  d  dxW(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l  1  (x) − y2  (L)  �(ˆa + 2)x − ˆa2�  W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l  1  (x)  = 4hˆaxy2  (L)  d  dxW(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−1  1  (x) + 2hˆa  �  x2 − 3(ˆa + 2)x + 2ˆa2�  W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−1  1  (x)  + 5  2x3y2  (L)  d3  dx3W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x) + x2 �  8x2 − 19(ˆa + 2)x + 11ˆa2� d2  dx2W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  + x  �  2x2 − 6(ˆa + 2)x + 5ˆa2� d  dxW(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x) + 2  �(ˆa + 2)x − ˆa2�  W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  − hˆax  �  5x2 d3  dx3 + 22x d2  dx2 + 14 d  dx  �  W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−3  1  (x)  −  �  x5 d5  dx5 + 10x4 d4  dx4 + 22x3 d3  dx3 + 4x2 d2  dx2 − 4x d  dx  �  W(L),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−4  1  (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  where we set W(L),−1  1  := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Before moving onto the Cauchy ensembles, we present the differential equations char-  acterising the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) in the JUE case with a = ˆaκN, b = ˆbκN as discussed  immediately prior to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do not display the analogous results in the JOE  and JSE cases due to their (relatively speaking) unwieldy forms, but note that they can be  derived from equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) following the same proof as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let y(J) :=  �  (ˆa + ˆb + 2)2x2 − 2(ˆa + 2)(ˆa + ˆb)x + ˆa2 with ˆa, ˆb constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The expansion coefficients specified by equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) in the JUE case with a = ˆaκN and b = ˆbκN  satisfy the differential equation  x(x − 1)y2  (J)  d  dxW(J),0  1  (x)  −  �  ˆa2(1 − x)3 − ˆa(ˆb + 2)x(1 − x)(1 − 2x) − (ˆb + 2)2x3 + 2(ˆb + 1)(3x2 + x)  �  W(J),0  1  (x)  = (ˆa + ˆb + 1)(ˆa(1 − x) + ˆbx),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  and for l ⩾ 2, the general differential equation  x(1 − x)y2  (J)  d  dxW(J),l  1  (x)  +  �  ˆa2(1 − x)3 − ˆa(ˆb + 2)x(1 − x)(1 − 2x) − (ˆb + 2)2x3 + 2(ˆb + 1)(3x2 + x)  �  W(J),l  1  (x)  = x3(1 − x)3 d3  dx3W(J),l−2  1  (x) + 4x2(1 − x)2(1 − 2x) d2  dx2W(J),l−2  1  (x)  − 2x(1 − x)(7x(1 − x) − 1) d  dxW(J),l−2  1  (x) − 2x(1 − x)(1 − 2x)W(J),l−2  1  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substitute expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) into equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and set a = ˆaN, b = ˆbN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equating  terms of equal order in N then produces the sought differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Similar to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13, in the β = 2, a = ˆaN, b = ˆbN setting, W(J),k  1  (x) vanishes for  odd k, and the differential equations of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15 constitute a recursion over even l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Differential equations for W(Cy),l  1  (x) in the symmetric case α real  The procedure outlined above extends to the Cauchy ensembles in the expected manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Having already set α = ˆακN (recall that η = −κ(N − 1) − 1 − α) to ensure that the global  scaled eigenvalue density ˜ρ(Cy)(λ) has the necessary properties, we simplify further by  constraining ˆα to be a real and positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This decision to consider only the symmetric  Cauchy ensembles is simply due to the fact that the equivalent results in the non-symmetric  case are cumbersome to display;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' they can however be derived in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, setting  α = ˆακN in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) with ˆα = O(1) in N, substituting in the expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  for ˜W(Cy)  1  (x), and then collecting terms of like order in N, we obtain the following two  propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Cauchy weight, set η = −κ(N − 1) − 1 − α, α = ˆακN with ˆα positive  real and constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, let y(Cy) :=  √  ˆα2x2 − 2ˆα − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the expansion coefficients of  the symmetric CyUE scaled resolvent ˜W(Cy)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2)  ��  ˆα∈R+ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) satisfy the differential equation  − (1 + x2)y2  (Cy)  d  dxW(Cy),0  1  (x) −  �ˆα2x2 − (ˆα + 2)2 + 2  �  xW(Cy),0  1  (x) = (ˆα + 1)(2ˆα + 1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  105  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  and for l ⩾ 2, the general differential equation  4(1 + x2)y2  (Cy)  d  dxW(Cy),l  1  (x) + 4  �ˆα2x2 − (ˆα + 2)2 + 2  �  xW(Cy),l  1  (x)  = (1 + x2)3 d3  dx3W(Cy),l−2  1  (x) + 8x(1 + x2)2 d2  dx2W(Cy),l−2  1  (x)  + 2(7x2 + 3)(1 + x2) d  dxW(Cy),l−2  1  (x) + 4x(1 + x2)W(Cy),l−2  1  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  As with the other classical unitary ensembles, ˜W(Cy)  1  (x) is an odd function of N when  β = 2, so W(Cy),k  1  (x) = 0 when k is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It can thus be easily checked that differential-  difference equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) trivially holds true for odd values of l and is otherwise to be  interpreted as a recursion over even l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Retain the notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 1 and 4, the expansion coefficients  of the scaled resolvent ˜W(Cy)  1  (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β)  ��  ˆα∈R+ satisfy the differential equations  − (1 + x2)y2  (Cy)  d  dxW(Cy),0  1  (x) −  �ˆα2x2 − (ˆα + 2)2 + 2  �  xW(Cy),0  1  (x) = (ˆα + 1)(2ˆα + 1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  2(1 + x2)y4  (Cy)  d  dxW(Cy),1  1  (x) + 2y2  (Cy)  �ˆα2x2 − (ˆα + 2)2 + 2  �  xW(Cy),1  1  (x)  = 4(κ − 1)ˆαy2  (Cy)(1+ x2)2 d  dxW(Cy),0  1  (x) + 2(κ − 1)ˆα  �  2ˆα2x2 − (ˆα + 3)2 + 6  �  x(1+ x2)W(Cy),0  1  (x)  + (κ − 1)ˆα  �(8ˆα2 + 9ˆα + 2)x2 + (2ˆα + 1)(5ˆα + 4)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  8(1 + x2)y4  (Cy)  d  dxW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  1  (x) + 8y2  (Cy)  �ˆα2x2 − (ˆα + 2)2 + 2  �  xW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  1  (x)  = 16(κ − 1)ˆαy2  (Cy)(1 + x2)2 d  dxW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  1  (x)  + 8(κ − 1)ˆα  �  2ˆα2x2 − (ˆα + 3)2 + 6  �  x(1+ x2)W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  1  (x) + 10(κ − 1)2y2  (Cy)(1+ x2)3 d3  dx3W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  1  (x)  + 4(κ − 1)2 �  19ˆα2x2 − 3ˆα2 − 22(2ˆα + 1)  �  x(1 + x2)2 d2  dx2W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  1  (x)  + 4(κ − 1)2 �  29ˆα2x4 − 2ˆα2x2 − 44(2ˆα + 1)x2 + ˆα2 − 12(2ˆα + 1)  �  (1 + x2) d  dxW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  1  (x)  + 8(κ − 1)2 �  3ˆα2x4 − (ˆα + 8)2x2 − 4(2ˆα + 1)  �  W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  1  (x)  − 4(κ − 1)2 �(3ˆα2 − 1)x2 + 9ˆα2 + 10ˆα + 3  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  106  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  and for l ⩾ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the general differential equation  4(1 + x2)y4  (Cy)  d  dxW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l  1  (x) + 4y2  (Cy)  �ˆα2x2 − (ˆα + 2)2 + 2  �  xW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l  1  (x)  = 8(κ − 1)ˆαy2  (Cy)(1 + x2)2 d  dxW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−1  1  (x)  + 4(κ − 1)ˆα  �  2ˆα2x2 − (ˆα + 3)2 + 6  �  x(1 + x2)W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−1  1  (x)  + 5(κ − 1)2y2  (Cy)(1 + x2)3 d3  dx3W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  + 2(κ − 1)2 �  19ˆα2x2 − 3ˆα2 − 22(2ˆα + 1)  �  x(1 + x2)2 d2  dx2W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  + 2(κ − 1)2 �  29ˆα2x4 − 2ˆα2x2 − 44(2ˆα + 1)x2 + ˆα2 − 12(2ˆα + 1)  �  (1 + x2) d  dxW(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  + 4(κ − 1)2 �  3ˆα2x4 − (ˆα + 8)2x2 − 4(2ˆα + 1)  �  W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−2  1  (x)  − (κ − 1)3ˆα  �  5(1 + x2)4 d3  dx3 + 38x(1 + x2)3 d2  dx2  +2(31x2 + 15)(1 + x2)2 d  dx + 4x(4x4 + 9x2 + 5)  �  W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−3  1  (x)  − (κ − 1)4  �  (1 + x2)5 d5  dx5 + 20x(1 + x2)4 d4  dx4 + (122x2 + 29)(1 + x2)3 d3  dx3  + 4x(65x2 + 46)(1 + x2)2 d2  dx2 + 4(41x4 + 56x2 + 14)(1 + x2) d  dx  + 8x(2x4 + 4x2 + 3)  �  W(Cy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='l−4  1  (x) + 4(κ − 1)3ˆαx2χl=3 + 2(κ − 1)4(1 − x2)χl=4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  where we have set W(Cy),−1  1  := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It can be observed that throughout this subsection, the differential equa-  tions characterising the leading order terms W0  1(x) depend on the classical weight  w(λ), but are otherwise the same for all β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, W0  1(x) is β-independent and use  of the Sokhotski–Plemelj inversion formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) tells us that the large N limiting  form ρ0(λ) of ˜ρ(λ) also has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, with a, b = O(N),  W1  1(x) vanishes for β = 2 but can be seen to be non-zero for β ̸= 2, in keeping with  Note 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 following Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, solving the differential equations of Proposi-  tions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15 with the boundary conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) recovers expressions for  W0  1(x) known from the general-β works [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying the Sokhotski–Plemelj  formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) then recovers the expressions for ρ0(λ) given in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and  its 1/N, 1/N2 corrections seen in [319], [142] (see also the earlier work [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  107  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' According to the second point of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9 (more specifically equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='88)),  taking a = ˆaκN and b = ˆbκN with ˆa, ˆb constant in N is equivalent to setting γ1 = 1 + ˆa  and γ2 = 1 + ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking this into account, we see that the auxiliary functions y(w) used  in Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 are related to the large N limiting forms given in Propo-  sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be precise, y(G), y(L), y(J), and y(Cy) are equal to the Stieltjes transform  of ρ(G),0(λ), 2λρ(L),0(λ), 2λ(1 − λ)ρ(J),0(λ), and (1 + λ2)ρ(Cy),0(λ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  significance of their presence in the differential equations of this subsection is that for  l > 0, the singularities of Wl  1(x) lie exactly at the zeroes of y(w), a fact well known from  the viewpoint of topological recursion [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We do not make explicit here the expressions for Wl  1(x) nor ρl(x) in the Gaussian,  Laguerre, or Jacobi cases, since these can be found in the literature [17], [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead,  we demonstrate the structures discussed in the above remark in the Cauchy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Retain the notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16, with ˆα positive real and constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For β = 1, 2, and 4, we have  W(Cy),0  1  (x) = (ˆα + 1)x − y(Cy)  1 + x2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  W(Cy),1  1  (x) = (κ − 1)ˆα  2  �  1  y(Cy)  −  ˆαx  y2  (Cy)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  W(Cy),2  1  (x) = (κ − 1)2ˆα  2  �  ˆα2((4ˆα2 + 10ˆα + 5)x2 + 2ˆα + 1)  4y5  (Cy)  − ˆα(ˆα2 + 2ˆα + 1)x  y4  (Cy)  �  + κ  8  ˆα2(2ˆα + 1)(1 + x2)  y5  (Cy)  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  along with the large N asymptotic for the smoothed eigenvalue density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  ˜ρ(Cy)(λ)  ���  ˆα∈R+  = ρ(Cy),0(λ) + κ − 1  κN  �  ˆα  2π  √  1 + 2ˆα − ˆα2λ2 χ|λ|<√  2ˆα+1/ˆα  −1  4δ(λ −  √  2ˆα + 1/ˆα) − 1  4δ(λ +  √  2ˆα + 1/ˆα)  �  + O  � 1  N2  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  where δ(λ) is the Dirac delta and ρ(Cy),0(λ) is specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) with ˆα = ˆα1 and ˆα2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The general solution of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29), equivalently (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31), is  W(Cy),0  1  (x) = (ˆα + 1)x + Cy(Cy)  1 + x2  ,  108  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  with the integration constant C forced to equal −1 due to the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Similarly, the integration constants present in the general solutions of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) must be zero due to the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) is  obtained by applying the Sokhotski–Plemelj formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) to the series (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) truncated  to have two terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' note that the Stieltjes transform (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) of δ(λ − λ0) is 1/(x − λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Before moving on to the soft and hard edge scaling regimes, let us remark that the results  contained in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18 are expected to hold for general real β > 0, in line with what  is known about the other classical β ensembles [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Soft and hard edge scaled differential equations  In this subsection we study the eigenvalue densities of the classical matrix ensembles after  they have been shifted to be centred on either the largest or smallest eigenvalue and scaled  such that the mean spacing between this eigenvalue and its neighbour is order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  large N limits of the eigenvalue densities recentred and scaled in this manner have one of  two universal forms, depending on whether the centring is performed at a soft or hard edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  we denote these large N limiting forms ρ(so f t)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) and ρ(hard)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling  from the discussion following Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, centring on the largest (smallest) eigenvalue  corresponds to a soft edge if ρ0(λ) exhibits a square root profile at the upper (lower) endpoint  of its support, and a hard edge if it instead has an inverse square root profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, some  observations can be made from the contents of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 using equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='88): When  the parameters a, b, α are proportional to N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', upon setting a = ˆaκN, b = ˆbκN, and  α = ˆακN with ˆa, ˆb, ˆα constant in N, all of the edges of the classical β ensembles are of the soft  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If one instead takes a = O(1), the lower edge of the Laguerre and Jacobi β ensembles  is of the hard type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, if b = O(1), ρ(J),0(λ) has a hard edge at the upper endpoint  of its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A point of interest we will observe is that while edge scaling is a procedure  local to the endpoint of support being centred on, it depends on both parameters a, b where  applicable, even though a (b) mainly governs the behaviour of the lower (upper) edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The particular scalings required to ensure that the large N limits of the eigenvalue  densities of the classical matrix ensembles share the same universal forms can be a little  complicated, so we (re)introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall from the definitions of a, b, γ1, γ2 in  109  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='88) contained in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9 that  γ1 = lim  N→∞  M1  N = 1 + lim  N→∞  a − κ + 1  κN  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  γ2 = lim  N→∞  M2  N = 1 + lim  N→∞  b − κ + 1  κN  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  where M1, M2 are now interpreted as continuous parameters dependent on a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus in  the large N limit, γ1, γ2 = 1 when a, b = O(1), while if we set a = ˆaκN and b = ˆbκN  with ˆa, ˆb = O(1), we have γ1 = 1 + ˆa and γ2 = 1 + ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall too from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  that λ(MP)  ±  , λ(Wac)  ±  , λ(Cy)  ±  denote the endpoints of the supports of ˜ρ(L)(λ), ˜ρ(J)(λ), ˜ρ(Cy)(λ),  respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in the following, we only consider the symmetric Cauchy ensembles, so we set  ˆα ∈ R and λ(Cy)  ±  = ±√  2ˆα + 1/ˆα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us also define  γ3 := (ˆα + 1)2  ˆα3  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  for convenience and  q2 :=  M1  M1 + M2  ,  r2 :=  M2  M1 + M2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  ˜q2 :=  N  M1 + M2  ,  ˜r2 := 1 − ˜q2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43)  uN :=  �  qr ˜q˜r√M1 + M2  ˜q˜r(q2 − r2) + qr( ˜q2 − ˜r2)  �4/3  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  following [178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Differential equations characterising the soft edge  We now present differential equations satisfied by ρ(so f t)(λ) for β ∈ {2/3, 1, 2, 4, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define the soft edge limiting forms of the differential operators DN,β introduced in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 as  D(so f t)  β  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  d3  dx3 − 4x d  dx + 2,  β = 2,  d5  dx5 − 10κx d3  dx3 + 6κ d2  dx2 + 16κ2x2 d  dx − 8κ2x,  β = 1, 4,  3 d7  dx7 − 56κx d5  dx5 + 28κ d4  dx4 + 784  3 κ2x2 d3  dx3  −208κ2x d2  dx2 − 4κ2(64κx3 − 17) d  dx + 128κ3x2,  β = 2/3, 6,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45)  110  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  where we recall that κ = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for β ∈ {2/3, 1, 2, 4, 6}, the Gaussian, Laguerre, Jacobi, and  symmetric Cauchy β ensembles’ eigenvalue densities satisfy the following differential equation in the  soft edge limit:  D(so f t)  β  ρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Make the change of variables [115] x �→ √κ  �√  2N +  x  √  2N1/6  �  in equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, multiply through by N−1/2 for β = 2, N−5/6 for β = 1 and 4, or N−7/6 for  β = 2/3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equating terms of order one then yields equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) above, while all  other terms vanish in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the case β = 2, the differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) satisfied by the soft edge density has  been isolated in the earlier works of Brack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [50, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) with ¯h2/m = 1, λM − V(r) =  − 1  2r, D = 1] and of Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [77, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (207) with d = 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 1, 2, and 4, the above proof  can be replicated by instead considering the differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) for the eigenvalue densities of the Jacobi, symmetric Cauchy, and Laguerre  ensembles, due to universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Explicitly, differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) is satisfied by the  leading order term of the following scaled densities in the large N limit:  In the regime of the largest eigenvalue [115], [28], [132], [133],  ρ(G)  �√  κ  �√  2N + δ(G) +  x  √  2N1/6  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  where δ(G) = o(N−1/6) is an arbitrary parameter (the regime of the smallest eigenvalue  can be treated by exploiting the symmetry x ↔ −x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the regime of the largest eigenvalue [115], [196], [121], [132], [133],  ρ(L)  �  κ  �  λ(MP)  +  N + δ(L,so f t)  +  + (√γ1 + 1)4/3  √γ1  N1/3x  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48)  where δ(L,so f t)  ±  = o(N1/3) is henceforth an arbitrary parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that this choice of  scaling is effective whether we set a = ˆaκN or take a to be constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the latter  case, the argument of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48) simplifies to κ(4N + δ(L,so f t)  +  + 24/3N1/3x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the regime of the smallest eigenvalue, having set a = ˆaκN with ˆa = O(1) [28], [121],  ρ(L)  �  κ  �  λ(MP)  −  N − δ(L,so f t)  −  − (√γ1 − 1)4/3  � N  √γ1  �1/3  x  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49)  111  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  In the regime of the largest eigenvalue, with b = ˆbκN and ˆb = O(1) [197], [121], [178],  ρ(J)  �  λ(Wac)  +  + δ(J,so f t) + qr ˜q˜r  uN  x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50)  where δ(J,so f t) = o(N−2/3) is arbitrary and q, r, ˜q, ˜r, uN are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Like with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48), this scaling is effective regardless of whether a is constant or linear  in N (we must take b = O(N) to ensure that this is actually a soft edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The regime of  the smallest eigenvalue can be treated by using the symmetry (x, a, b) ↔ (1 − x, b, a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the regime of the largest eigenvalue, having set α = ˆακN with ˆα positive real and  constant in N,  ρ(Cy)  �  �λ(Cy)  +  + δ(Cy) +  �  γ2  3  2λ(Cy)  +  N2  �1/3  x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β  �  � ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)  where δ(Cy) = o(N−2/3) is an arbitrary parameter and γ3 = (ˆα + 1)2/ˆα3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For even β, ρ(so f t)(x) has an explicit representation as a β-dimensional integral  due to [81], while [140] provides alternate forms for β = 1, 2, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To date, there is no  explicit functional form for ρ(so f t)(x) when β = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, [90] shows for all  κ = β/2 > 0 that for the first two leading orders as x → ∞,  ρ(so f t)(x) ∝ exp  �−4κx3/2/3  �  x3κ/2  ,  which is an extension of the even-β result of [120],  ρ(so f t)(x) ∼  x→∞  1  π  Γ(1 + κ)  (8κ)κ  exp  �−4κx3/2/3  �  x3κ/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  This result is consistent with the differential equations of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' So too is the result  [81],  ρ(so f t)(x)  ∼  x→−∞  �  |x|  π  ,  β ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53)  Likewise, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 below is consistent with the result [116],  ρ(hard)(x) ∼  x→∞  1  2π√x,  β ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  Note that the asymptotic forms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54) respectively capture the facts  that moving past the soft edge results in exponential decay, moving from the soft edge into  the bulk results in a square root profile, and moving from the hard edge into the bulk shows  a square root singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  112  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  Differential equations characterising the hard edge  We now give the hard edge analogue of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 for β ∈ {1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For uniformity across  the Laguerre and Jacobi ensembles, we study the lower edge centred at x = 0 (the hard edge  at x = 1 for the Jacobi ensemble can be studied by interchanging x with 1 − x and a with b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In contrast to the soft edge scaling, the differential equations characterising the hard edge  are seen to depend on the a paramater, which is taken to be constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define the hard edge limiting forms of the differential operators DN,β specified in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 as  D(hard)  β  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  x3 d3  dx3 + 4x2 d2  dx2 +  �  x − a2 + 2  �  x d  dx + 1  2x − a2,  β = 2,  4x5 d5  dx5 + 40x4 d4  dx4 + [10κx − 5˜a + 88] x3 d3  dx3  + [38κx − 22˜a + 16] x2 d2  dx2  +  �  (2κx − ˜a)2 + 12κx − 14˜a − 16  �  x d  dx  +(2κx − ˜a)(κx − ˜a) − 4κx,  β = 1, 4,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='55)  where we retain the definition of ˜a given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for β ∈ {1, 2, 4}, the hard edge scaled  eigenvalue densities of the Laguerre and Jacobi β ensembles satisfy the differential equation  D(hard)  β  ρ(hard)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since the Gaussian ensembles do not exhibit a hard edge, we turn to the Laguerre  ensembles as they are the next simplest to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we begin by changing variables  [115], [256] x �→ κx/(4N) in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equating terms of order one yields the  differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56) above, and we note that all other terms are O( 1  N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As in the soft edge case, there is a little bit of freedom in the choice of scaling used in  the above proof, and the proof itself can be formulated in terms of the differential equations  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) characterising the eigenvalue densities of the Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be  precise, the differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56) above is satisfied by the leading order term of the  following scaled densities in the large N limit:  With δ(L,hard) = o(N) an arbitrary parameter [115], [256], [134],  ρ(L)  �  κx  4N + δ(L,hard) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57)  113  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  With δ(J,hard) = o(N) also an arbitrary parameter,  ρ(J)  �  x  N(4γ2N + δ(J,hard));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58)  While a must be constant in N, applying this scaling to the differential equations of  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 recovers Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 regardless of whether we take b to be proportional to  N or constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Optimal scaling at the soft and hard edges  It has previously been observed in [132] that the differential equation characterisation of the  soft edge scaled eigenvalue density can be extended to similarly characterise the optimal  leading order correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The latter is obtained by a tuning of the δ( · ) parameters in  equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) so as to obtain the fastest possible decay in N of the leading order  correction, and thus the fastest possible convergence to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On this latter point, and  considering the Gaussian case for definiteness, we know from [133] that for β even (at least),  ˆρN,β(x) :=  √κ  √  2N7/6 ρ(G)  �√  κ  �√  2N + δ(G) +  x  √  2N1/6  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='59)  = ρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) +  �√  2N1/6δ(G) − (1 − 1/κ)/(2N1/3)  � d  dxρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β)  + O(N−2/3),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='60)  where the O(N−2/3) terms do not depend as simply on ρ(so f t)(x), d  dxρ(so f t)(x) as those made  explicit above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, choosing δ(G) = (1 − 1/κ)/(2  √  2N) gives the fastest convergence to  the limit,  ˆρN,β(x) = ρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) +  1  N2/3 ζβ(x) + o(N−2/3)  for some ζβ(x) which, for the values of β permitting a differential equation characterisation  of ρ(G)(x) and ρ(so f t)(x), can itself be characterised as the solution of a differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The simplest case is β = 2, when  ζ′′′  2 (x) − 4xζ′  2(x) + 2ζ2(x) = x2 d  dxρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2) − xρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61)  which is an inhomogeneous generalisation of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) for β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For the particular  Laguerre soft edge scaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48), again considering β = 2, the analogue of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61)  is given in [132, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)], with a = ˆaκN and δ(L,so f t)  +  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  114  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Scalings of the Differential Equations  More generally, multiplying the functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) by appropriate scalars and  powers of N results in a function whose large N asymptotic expansion has leading order  term given by ρ(so f t)(x) and next to leading order correction some function generically  O(N−1/3) (when β = 2 and not all relevant parameters are constant, the correction term  is actually of order N−2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, multiplying the functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58) by  the correct pre-factors produces a function whose large N limit is given by ρ(hard)(x) with  correction term of generic order 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In a similar fashion to the computation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='60) above,  proper tuning of the δ( · ) parameters seen in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58) ensures that  the correction terms are O(N−2/3) in the soft edge case, and O(1/N2) in the hard edge  case, which corresponds to the scaled eigenvalue densities experiencing the optimal rate of  convergence to the limiting forms ρ(so f t)(x) and ρ(hard)(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For β an even integer, [133] shows that the scaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48) becomes optimal when we set  δ(L,so f t)  +  =  �  �  �  �  �  2a  κ ,  a = O(1),  �  1 − 1  κ  � γ1−1  2√γ1 ,  a = O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='62)  Likewise, [134] shows for general real β > 0 and a ∈ N that the scaling (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57) is optimal  upon setting δ(L,hard) = 2a/κ, which coincides with the value of δ(L,so f t)  +  given above for the  a = O(1) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We are able to extend these results by utilising Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  as follows: Letting ˆρN,β(x) denote the analogue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='59) constructed from either of the  functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57), or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58), we have that ˆρN,β(x) is, at leading order,  equal to either ρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) or ρ(hard)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, ˆρN,β(x) satisfies an edge-scaled version of  one of the differential equations of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, while ρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) and ρ(hard)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) satisfy  the differential equations of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, the correction term ˆρN,β(x) −  ρ(so f t)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β), respectively ˆρN,β(x) − ρ(hard)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β), can be seen to satisfy a certain differential  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Analysis of this latter differential equation reveals the order in N of the correction  term and its dependence on the δ( · ) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, for β = 1, 2, and 4, we are able to  identify the values of δ( · ) which optimally tune the scalings (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57),  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We confirm that the values δ(G) = (1 − 1/κ)/(2  √  2N), δ(L,so f t)  +  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='62), and  δ(L,hard) = 2a/κ recounted above from the works [132], [133], [134] remain optimal when  β = 1 and/or a > −1 is allowed to be general real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, we find that the scalings  115  CHAPTER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Differential Equations for the Classical Matrix Ensembles  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58) are optimal when taking  δ(L,so f t)  −  =  �  1 − 1  κ  � γ1 − 1  2√γ1  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63)  δ(Cy) =  �  1 − 1  κ  �  γ3  2λ(Cy)  +  N  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='64)  δ(J,hard) =  �  �  �  �  �  4(a+b)  κ  − 4  �  1 − 1  κ  �  ,  b = O(1),  2a  κ (γ2 + 1) − 4  �  1 − 1  κ  �  ,  b = O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65)  Note that the value of δ(L,so f t)  −  given above is equal to that of δ(L,so f t)  +  given in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='62)  in the a = O(N) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, the value of δ(J,hard) given above for the b = O(1) case  agrees with computations seen in the recent study [126, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Due to computational  limitations, we have not been able to identify the optimal value of δ(J,so f t) for β = 1, 4, but  note that its value in the β = 1 case can be surmised from [197] (which presumably extends  to the β = 4 case through the duality principles of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), while for β = 2, it should  reduce to zero in the same fashion as δ(G), δ(L,so f t)  −  , δ(Cy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  116  Chapter 3  Characterisations of the Moments and  Cumulants  It was mentioned in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that the eigenvalue densities ρ(λ) of the random matrix ensembles  studied in this thesis are fully characterised by their spectral moments (though, one must  keep in mind the caveat regarding the Cauchy ensembles)  mk =  �  R λkρ(λ) dλ =  �  N  ∑  i=1  λk  i  �  ,  k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  In the above, the latter presentation of the spectral moments is to be read as an average with  respect to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of the matrix ensemble of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We begin this chapter  with an application of the differential equations given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 to the derivation of  linear recurrences characterising the spectral moments of the classical matrix ensembles (and  the Gaussian β ensembles with β = 2/3, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We recall from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 that the classical  matrix ensembles are exactly those whose eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s are of the form  p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) =  1  N (w)  N,β  N  ∏  i=1  w(λi) |∆N(λ)|β,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  where β = 1, 2, or 4,  ∆N(λ) =  ∏  1⩽i<j⩽N  (λj − λi)  is the Vandermonde determinant (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), and w(λ) is a suitable classical weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular,  we are interested in the spectral moments of the Gaussian, Laguerre, Jacobi, symmetric  117  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  shifted Jacobi, and symmetric Cauchy ensembles, which are specified respectively by the  weights (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  w(G)(λ) = e−λ2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  w(L)(λ) = λae−λχλ>0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  w(J)(λ) = λa(1 − λ)bχ0<λ<1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  w(sJ)(λ)  ��  a=b = (1 − λ2)aχ−1<λ<1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  w(Cy)(λ)  ��  α∈R = (1 + λ2)−κ(N−1)−1−α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  we recall that κ := β/2 and the indicator function χA is defined to equal one when A is true  and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The parameters a, b, α are taken to be real throughout this chapter and,  for convergence issues, are further constrained such that a, b > −1 and α > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The aforementioned linear recurrences on the spectral moments are given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, where  we also show that these recurrence relations hold true for the negative-integer moments (m−k  with k ∈ N), which have gained interest primarily due to their connection to problems of  quantum transport [56], [240], [241], [71] and more recently due to duality principles relating  the negative-integer moments m−k to their counterparts mk−1 [72];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in fact, we show moreover  that our moment recurrences are satisfied by a further generalisation (see equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  forthcoming) of the mk concerning complex values of k, which have recently been studied  in [72], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Having linear recurrence relations on the spectral moments of the classical  matrix ensembles at hand means that we are able to derive so-called 1-point recursions [61]  characterising the moment expansion coefficients defined in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be precise, we  proceed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 by substituting the expansions  m(G)  2k =  k  ∑  l=0  M(G)  k,l N1+k−l,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  m(L)  k  =  k  ∑  l=0  M(L)  k,l N1+k−l,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  m(J)  k  =  ∞  ∑  l=0  M(J)  k,l N1−l  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  into the moment recurrences of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and equating terms of like order in N to obtain 1-point  recursions on the M(w)  k,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following [61, Defn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1], this means that we find for each ensemble  some integers imax, jmax and a non-trivial set of polynomials {q(w)  ij (k)}0⩽i⩽imax,0⩽j⩽jmax such  118  that whenever the involved terms are well-defined, we have the equality  imax  ∑  i=0  jmax  ∑  j=0  q(w)  ij (k)M(w)  k−i,l−j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  We present recursions of this type for the moment expansion coefficients of the Gaussian β  ensembles with β = 2/3, 6, the (classical) Laguerre ensembles, and the (symmetric shifted)  Jacobi unitary ensemble, choosing to forgo consideration of the remaining classical matrix  ensembles in the interest of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we confirm here that applying our method to  the GUE recovers the celebrated Harer–Zagier recursion [174]  4(k + 1)M(GUE)  k,l  = 4(2k − 1)M(GUE)  k−1,l + (k − 1)(2k − 1)(2k − 3)M(GUE)  k−2,l−2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  subject to the initial conditions M(GUE)  0,0  = 2M(GUE)  1,0  = 1, M(GUE)  1,1  = 0, and M(GUE)  k,l  = 0 for all  k, l < 0, whereas treating instead the GOE and GSE recovers the corresponding recursions of  Ledoux given in [224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Evidently, the problem of computing the spectral moments of the classical matrix en-  sembles is not a new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Aside from recurrences on the moments mk and their expansion  coefficients Mk,l known from the works [174], [169], [223], [224], [319], [71], [72], [23] (which  the computations of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 recover and/or supplement), the literature contains a range  of different approaches for studying the spectral moments of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we  review some of these alternative characterisations of the spectral moments of the classical  matrix ensembles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in short, we survey parts of the literature that show how the study of  these spectral moments relates to skew-orthogonal polynomial theory, symmetric function  theory, and hypergeometric orthogonal polynomials from the Askey scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  After the review of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, the remainder of this chapter focuses on elucidating how  the spectral moments mk, mixed moments (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  n  ∏  i=1  Tr Xki  �  P(X)  ,  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  and mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn of particular random matrix ensembles relate to the enumeration  of ribbon graphs satisfying certain constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Here, the term ‘ribbon graph’ needs to be  formally introduced and it is helpful to reiterate the definition of ‘mixed cumulants’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  mixed cumulants cκi are defined implicitly by the moment-cumulants relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  ∑  K⊢{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn} ∏  κi∈K  cκi,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  119  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  where we recall that K ⊢ {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn} is read as “K is a partition of {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn}”, which means  that K = {κi}m  i=1 for some 1 ⩽ m ⩽ n such that the disjoint union κ1 ⊔ · · · ⊔ κm is equal to  {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, K ⊢ {4, 6} if and only if K is equal to { {4, 6} } or { {4}, {6} }, so  m4,6 = c4,6 + c4 c6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As for ribbon graphs, we give here a pictorial definition that is well suited to our purposes  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [211], [250], [219] for alternative but near-equivalent definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting k ∈ N, a k-ribbon graph is a collection of n polygons (0 < n ⩽ 2k) with  labelled vertices and k rectangles, referred to as ribbons, satisfying the following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The number of edges of the polygons must sum to 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For each ribbon, designate a pair of opposite edges to be referred to as ends of the  ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, each of the 2k polygon edges must be identified, in a topological sense,  with exactly one ribbon-end, so that each ribbon connects two polygon edges together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The Euler genus of an orientable (non-orientable) connected ribbon graph is the smallest  ˜g ∈ N such that the ribbon graph can be drawn on a sphere with ˜g/2 handles ( ˜g cross-caps)  without intersecting with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the orientable case, it is double the usual genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since there are two ways to identify a pair of lines, depending on the relative  orientation of the lines, we allow our ribbons to have at most one M¨obius half-twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We  refer to the class of ribbon graphs with untwisted ribbons as ‘(globally) orientable’, while  the term ‘locally orientable’ refers to the scenario where ribbons are allowed to be M¨obius  half-twisted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the literature, ribbon graphs are usually restricted to be of the first type,  while the latter type are referred to as M¨obius graphs [251], [201], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1: A 3-ribbon graph constructed from a monogon, bigon, and triangle, with a grey  ribbon connecting the single edge of the monogon to the left edge of the bigon, a M¨obius  half-twisted pink ribbon connecting the right edge of the bigon to an edge of the triangle,  and a blue ribbon connecting the remaining two edges of the triangle together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  120  The connection between ribbon graphs and random matrix theory was first explored by  Br´ezin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in their 1978 work [54] (see also [36]), where they were strongly influenced by  ’t Hooft’s 1974 work [179] on the large N limit of gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Closely related ideas were  seen independently in [326], [304], [174] over the next decade, with Harer and Zagier’s 1986  work [174] being of particular interest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Technically speaking, Harer and Zagier did  not look directly at ribbon graphs, but rather showed and utilised the fact that 2kM(GUE)  k,l  equals the number of ways of identifying pairs of edges of a 2k-gon to form a compact  orientable surface of genus l/2 — of course, this is equal to the number of orientable genus  l/2 ribbon graphs that can be built from a single 2k-gon since the ribbons can be interpreted  as identifying the polygon edges at their ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In 1997, Goulden and Jackson [165] (see also  [166], [251], [201]) gave the GOE analogue of the above statement, which is that 4kM(GOE)  k,l  equals the number of locally orientable ribbon graphs of Euler genus l that can be built from  a single 2k-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Other related works from this time period include [271], [211], [190], [287].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we give a detailed review of how the spectral moments of the GUE and GOE  relate to the ribbon graphs described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we review the analogous  statements for the LUE [82] and LOE [57];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the upshot here is that M(LUE)  k,l  , M(LOE)  k,l  count  the same ribbon graphs as M(GUE)  k,l  , M(GOE)  k,l  , respectively, except that the vertices of the  underlying 2k-gons are alternately coloured black and red, and the edge identifications  induced by the ribbons respect this bicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our arguments, which flow from the Isserlis–  Wick theorem (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), have been shown to extend to the GSE and  LSE [251], [57], but we do not explore this fact in order to keep our discussion concise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  M(GSE)  k,l  , M(LSE)  k,l  can roughly be ascribed the same combinatorial meaning as M(GOE)  k,l  , M(LOE)  k,l  due to the β ↔ 4/β duality of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we extend the arguments of  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to produce ribbon graph interpretations for the spectral moments of the  Hermitised and antisymmetrised matrix product ensembles defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  One of our motivations for supplying the review contained in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 is that it allows us to  interpret the 1-point recursions of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 pertaining to the Laguerre ensembles as solving  combinatorial problems similar to that considered by Harer and Zagier — unfortunately, we  are currently unable to find analogous applications for the other recursions given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  due to complications that we outline in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned earlier, the discussion of  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 does not focus solely on the spectral moments of the random matrix ensembles  121  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  at hand, but also their mixed moments and cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, we show therein how  straightforward generalisations of the ribbon graphs described above, now constructed  from multiple polygons, relate to the mixed moments and cumulants of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We  show moreover that the relevant mixed moments and cumulants are polynomial in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A consequence of this is that, upon appropriate scaling, we are able to assume that the  connected n-point correlators Wn (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) of the Hermitised and antisymmetrised matrix  product ensembles have large N (actually N0) expansions of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) without  needing to check the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The existence of these large N expansions  is significant because they are crucial to the loop equation analysis presented in Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Recurrence Relations for the Moments of the Classical Matrix  Ensembles  We now derive the aforementioned linear recurrence relations on the spectral moments mk  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), and explain why they hold true when the mk are instead taken to be the complex-k  generalised moments (following Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [72])  mk :=  �  R |λ|kρ(λ) dλ,  k ∈ {k′ ∈ C | mk′ < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  Note that these generalised moments agree with the spectral moments defined by equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) for k ∈ N in the Jacobi and Laguerre cases, and for k ∈ 2N in the Gaussian, symmetric  shifted Jacobi, and symmetric Cauchy cases — in the latter set of cases, mk = 0 for k odd  according to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), but not equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since the eigenvalue densities studied  in this section are either symmetric about the origin or have support contained within  the positive real line, the generalised moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) can be interpreted as the spectral  moments (we reserve the adjective ‘spectral’ for the moments defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)) of  the constrained density ρ(λ)χλ>0, with a possible factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The study of the former type of spectral moments is motivated by the fact that they  fully characterise the eigenvalue densities of the classical matrix ensembles, as discussed in  §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Generalisation to negative integers k was first considered by Brouwer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [56] in the  Laguerre case, where they showed that the reciprocated eigenvalues of the Wigner–Smith  time delay matrix Q are distributed according to the Laguerre ensemble with a = κN so that  122  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  �  Tr Qk� = m(L)  −k |a=κN [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The significance of the Wigner–Smith matrix is that it relates to  the study of ballistic chaotic scattering, whereby its eigenvalues, referred to as proper delay  times, describe the amount of time incident particles spend in the ballistic cavity during a  scattering event, among other observables [33], [56], [70], [71], [229], [240], [241].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The more  recent extension to the complex-k moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) considered by Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [72] relates  to the study of so-called spectral zeta functions of random matrices X, ζX(k) := Tr (X†X)−k  (note that if we take X to be the infinite-dimensional deterministic matrix diag(1,  √  2,  √  3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ),  ζX(k) is then the Riemann zeta function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is shown in [72] that spectral zeta functions  satisfy a reflection symmetry across the line Re(k) = 1/2 (much like the Riemann zeta  function), which implies reciprocity laws between the moments m−k and mk−1 of the LUE  and JUE, for k ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 within §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we return to the study of spectral moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) with k ∈ N since then, it is  known (recall Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) that m(G)  k  , m(L)  k  are polynomial in N and m(J)  k  is a rational function  in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) are valid and we are able to derive 1-point recur-  sions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) characterising the moment expansion coefficients Mk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned earlier,  1-point recursions satisfied by these moment expansion coefficients have been studied in spe-  cial cases, with particular attention paid to their topological or combinatorial interpretations  [174], [224], [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These aspects are briefly discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, with the highlighted point  being that even though the moment expansion coefficients Mk,l may have combinatorial  interpretations in certain cases (fully detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), there are presently no such  interpretations for the 1-point recursions satisfied by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Recurrences for the spectral moments  One may obtain recurrences for the spectral moments mk of the classical matrix ensembles by  recalling from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that the resolvent W1(x) acts as a generating function for these moments,  presuming that the moments are well-defined for all k ∈ N or otherwise interpreting the  large x expansion of W1(x) in a formal sense:  W1(x) =  ∞  ∑  k=0  mk  xk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  Substituting this series into the differential equations for W1(x) given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and  then equating terms of equal order in x gives relations between the spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  123  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  equations obtained from terms of negative order in x give the upcoming recurrences on the  spectral moments, while the terms of order one and positive order in x give the first few  moments required to run the recursions (the latter are also available in earlier literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [142] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The recurrence relations derived from the above procedure enable the iterative com-  putation of the spectral moment mk for any positive integer k starting with knowledge of  m0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , mp for some small integer p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, these recurrences are actually valid for the  complex-k generalisations defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), so long as all involved moments mk are  such that in the Gaussian and symmetric shifted Jacobi cases, Re(k) > −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in the Jacobi and  Laguerre cases, Re(k) > −a − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and in the symmetric Cauchy case, −1 < Re(k) < 2α + 1  (this is a refinement of the constraint −1 < k < 2α1 + 1 discussed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To ascertain  the validity of our moment recurrences in these more general settings, we must provide an  alternative proof to that outlined in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, rather than the differential  equations for W1(x), we begin at the differential equations for the eigenvalue density ρ(x)  given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Multiplying both sides of these differential equations by |x|k and then  integrating over the support of the eigenvalue density produces a relation between terms of  the form �  supp ρ xp|x|k dn  dxn ρ(x) dx for positive integers p, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The goal then is to use integration  by parts to reduce these terms to the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is done in a similar fashion to what  is shown in Appendix B, with the notable exception being that the domain of integration  must be split at the origin due to the absolute value sign in the factor |x|k — in any case, all  boundary terms arising from integration by parts vanish, owing to our restrictions on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall the definitions of ˜a, ˜b, and ˜c given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2 and k ∈ C  such that Re(k) > 2 − a, the moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) of the eigenvalue density of the Jacobi ensemble satisfy  the third order linear recurrence  3  ∑  l=0  d(J)  2,l m(J)  k−l = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  where  d(J)  2,0 = k  �(a + b + 2N)2 − (k − 1)2�  ,  d(J)  2,1 = 3k3 − 11k2 − k  �  2(a + b + 2N)2 + a2 − b2 − 14  �  + 3(a + b)(a + 2N) + 6(N2 − 1),  124  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  d(J)  2,2 = (2k − 3) [2N(a + b + N) + ab] − (k − 2)  �  3k2 − 10k − 3a2 + 9  �  ,  d(J)  2,3 = (k − 3)  �(k − 2)2 − a2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For β = 1, 4 and k ∈ C such that Re(k) > 4 − a, the moments of the Jacobi ensemble’s eigenvalue  density satisfy the fifth order linear recurrence  5  ∑  l=0  d(J)  4,l m(J)  k−l = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  where  d(J)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0 = k(˜c2 − (k − 2)2)(˜c2 − (2k − 1)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(J)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 = 1  2(˜c2 − 9)2(5 − 6k) + 1  2(˜a − ˜b)  �(˜c2 − 9)(5 − 4k) + 2k(5(k − 1)(k − 5) + 4k)  �  + (˜c2 − 9)k [5(4k − 3)(k − 3) + 2k] − 4k2(k − 5) [5(k − 2)(k − 1) − 2] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(J)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 = ˜c4(3k − 5) + ˜c2 � 1  2(˜a + ˜b)(2k − 5) + 5(˜a − ˜b)(k − 2)  �  − ˜c2 �  30k3 − 171k2 + 339k − 230  � − 1  2(˜a + ˜b)  �  5k3 − 44k2 + 129k − 125  �  − 1  2(˜a − ˜b)  �  35k3 − 246k2 + 581k − 460  � + 1  2(2k − 5)(˜a − ˜b)2  + 40k4(k − 11) + 1966k3 − 4443k2 + 5056k − 2305,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(J)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 = 1  2 ˜c4(5 − 2k) + ˜c2 � 1  4(˜a + ˜b)(25 − 8k) + 1  4(˜a − ˜b)(45 − 16k)  �  + ˜c2 �  20k3 − 155k2 + 401k − 345  � + 5  4(˜a + ˜b)  �  6k3 − 62k2 + 216k − 253  �  + 1  4(˜a − ˜b)  �  90k3 − 806k2 + 2436k − 2485  � + 1  4(˜a2 − ˜b2)(15 − 4k)  + 1  4(˜a − ˜b)2(25 − 8k) − 4k3(10k2 − 140k + 789) + 8923k2 − 12600k + 14125  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(J)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 = (k − 4)  �  k3 + k2 − 18k −  �˜c2 − ˜b − 4k2 + 29k − 51  � �  5k2 − 29k + 40  ��  + 1  2 ˜a2(6k − 25) + 1  2 ˜a  �(4k − 15)  �˜c2 − ˜b − 10k2 + 76k − 147  � − 2(k − 5)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(J)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 = (k − 5) [4(k − 5)(k − 4) − ˜a] [˜a − (k − 4)(k − 2)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The sequences of spectral moments {m(J)  k }k∈Z, k>−a−1 are fully determined for β = 1, 2, 4 by the above  recurrence relations and the initial terms m(J)  0 , m(J)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m(J)  4 , which can be computed through MOPS  [96] or via the methods presented in [242], [142] — according to the discussion at the beginning of  this subsection, the required initial terms are also encoded within the right-hand sides of differential  equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  125  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  The recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) for the moments m(J)  k  of the JUE eigenvalue density was recently  given in [72, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8], wherein it is formulated as a recurrence on the differences of moments  ∆m(J)  k  := m(J)  k  − m(J)  k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The fact that this recurrence relation can be naturally formulated  in terms of the differences ∆m(J)  k  is equivalent to the observation that ∑3  l=0 d(J)  2,l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since  ∑5  l=0 d(J)  4,l = 0, we see that it is also reasonable to rewrite the recurrence relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) in  terms of the ∆m(J)  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that the analogous recurrences for the symmetric shifted  Jacobi and symmetric Cauchy ensembles are most compactly presented when written in  terms of the differences and sums, respectively, of even moments  µ(sJ)  k  := m(sJ)  2k+2 − m(sJ)  2k ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  µ(Cy)  k  := m(Cy)  2k+2 + m(Cy)  2k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  Here, the moments mk are again defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) with ρ(λ) being the eigenvalue  density of the classical matrix ensemble specified by either the constrained weight (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) or  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The fact that the µ(sJ)  k  , µ(Cy)  k  recurrences are simpler than the correspond-  ing recurrences on the m(sJ)  2k , m(Cy)  2k  is in keeping with the fact that the former are, respectively,  the moments of r(sJ)(x) := (1 − x2)ρ(sJ)(x)|a=b and r(Cy)(x) := (1 + x2)ρ(Cy)(x)|α∈R, which  are characterised by differential equations whose coefficients are of degree two less than  the corresponding coefficients of the differential equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) for ρ(sJ)(x) and  ρ(Cy)(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the differential equations satisfied by r(sJ)(x) can be surmised from those satisfied  by r(Cy)(x), which are themselves given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 as equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define µ(sJ)  k  by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) and retain the definitions of aβ, ˜a, and ˜c given  in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2 and k ∈ C such that Re(k) > 1/2, we have the second order linear  recurrence  (2k + 4)  �(2k + 3)2 − 4(a + N)2�  µ(sJ)  k+1 − 2(2k + 1)  �(2k + 2)2 − 2N(N + 2a)  �  µ(sJ)  k  + (2k + 1)(2k)(2k − 1)µ(sJ)  k−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  The initial condition determining the sequence {µ(sJ)  k  |β=2}k∈N is  µ(sJ)  0  = 2N(a + N)(2a + N)  1 − 4(a + N)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  For β = 1, 4 and k ∈ C such that Re(k) > 3/2, we have the fourth order linear recurrence  2  ∑  l=−2  d(sJ)  4,l µ(sJ)  k−l = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  126  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  where  d(sJ)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−2 := −4(2k + 1)(2k)(2k − 1)(2k − 2)(2k − 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(sJ)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1 := (2k + 1)(2k)(2k − 1)  �  20˜a − 5˜c2 + 16k(4k + 5) + 77  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(sJ)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0 := (2k + 1)  �  8˜a(5k + 8)(2k + 3) −  �˜c2 − 4˜a − 30k2 − 67k − 42  �2  + 516k4 + 2292k3 + 3653k2 + 2534k + 673  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(sJ)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 := ˜c2 �(˜c2 − 4˜a)(4k + 7) − 120k3 − 656k2 − 1210k − 746  �  + ˜a  �  160k3 + 736k2 + 1128k + 580  �  + 512k5 + 4480k4 + 16056k3 + 29360k2 + 27270k + 10243,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  d(sJ)  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 := −2(k + 3)(˜c + 4k + 11)(˜c + 2k + 4)(˜c − 2k − 4)(˜c − 4k − 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The initial conditions determining the sequences {µ(sJ)  k  |β=1,4}k∈N are  µ(sJ)  0  = (˜c − 1)  �˜c + 2aβ − 3  � �˜c − 2aβ + 1  �  8˜c(˜c − 3)(1 − κ)  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  µ(sJ)  1  = (˜c2 − 5)(˜c − 7) − 4˜a(˜c − 1)  4(˜c2 − 4)(˜c − 7)  µ(sJ)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  We note that the generalisation of recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) concerning the non-symmetric shifted  JUE corresponding to the weight w(sJ)(λ) = (1 − λ)a(1 + λ)bχ−1<λ<1 and β = 2 was first  given by Ledoux in [223], albeit consideration was given only to the µ(sJ)  k  with k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moving  on, let us observe that the contents of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, particularly Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, imply the  relation  µ(sJ)  k  ���  a�→−κ(N−1)−1−α = (−1)k−1µ(Cy)  k  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  where we recall that κ = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A simple consequence of this relation is that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  leads to the following recurrence relations for the sums of moments µ(Cy)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Define µ(Cy)  k  by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) and retain the definitions of d(sJ)  4,−2, d(sJ)  4,−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , d(sJ)  4,2  listed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2 and k ∈ C such that 1/2 < Re(k) < α − 3/2, we have the  second order linear recurrence (recently given in [23])  (2k + 4)  �(2k + 3)2 − 4α2�  µ(Cy)  k+1 + 2(2k + 1)  �(2k + 2)2 + 2N(N + 2α)  �  µ(Cy)  k  + (2k + 1)(2k)(2k − 1)µ(Cy)  k−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  127  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  The initial condition determining the sequence {µ(Cy)  k  |β=2}k∈N is  µ(Cy)  0  =  2Nα(N + 2α)  (2α − 1)(2α + 1),  α > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  For β = 1, 4 and k ∈ C such that 3/2 < Re(k) < α − 5/2, we have the fourth order linear recurrence  2  ∑  l=−2  d(Cy)  4,l  µ(Cy)  k+l = 0,  d(Cy)  4,l  := (−1)l−1d(sJ)  4,l  ���  a�→−κ(N−1)−1−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  The initial conditions required to fully characterise the sequences {µ(Cy)  k  |β=1,4}k∈N are obtained by  parsing equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) through the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In a similar vein to how the arguments of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 enable us to simply translate the  recurrences of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 for the µ(sJ)  k  into the analogous recurrences on the µ(Cy)  k  given  in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, the limiting procedure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) described in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 can be used to obtain  recurrences on the (generalised) spectral moments m(L)  k  of the Laguerre ensembles from the  recurrences presented in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, to derive the following recurrences on the  m(L)  k , one may circumvent the strategy outlined at the beginning of this subsection by setting  m(J)  k  = b−km(L)  k  in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and then taking the limit b → ∞ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40), in  addition to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall the definitions of aβ, Nβ, and ˜a given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2 and k ∈ C  such that Re(k) > 1 − a, the moments of the LUE eigenvalue density satisfy the second order linear  recurrence  (k + 1)m(L)  k  = (2k − 1)(a + 2N)m(L)  k−1 + (k − 2)  �(k − 1)2 − a2�  m(L)  k−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  For β = 1, 4 and k ∈ C such that Re(k) > 3 − a, we have the fourth order linear recurrence  4  ∑  l=0  d(L)  4,l (κ − 1)lm(L)  k−l = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  where  d(L)  4,0 = k + 1,  d(L)  4,1 = (1 − 4k)(aβ + 4Nβ),  d(L)  4,2 = (1 − k)(5k2 − 11k + 4) + (2k − 3)  �˜a + 2(aβ + 4Nβ)2�  ,  d(L)  4,3 = (aβ + 4Nβ)  �(11 − 4k)˜a + 10k3 − 68k2 + 146k − 96  �  ,  d(L)  4,4 = (k − 4) [˜a − 4(k − 4)(k − 3)] [˜a − (k − 3)(k − 1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  128  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  The initial terms m(L)  0 , m(L)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m(L)  3  required to fully determine the sequences {m(L)  k }k∈Z, k>−a−1  in the orthogonal, unitary, and symplectic cases can be found in [242], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It should come as no surprise that recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) on the moments of the LUE  eigenvalue density has already been given in the literature, seeing as this is true of its JUE  analogue (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), which is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, it was first obtained by Haagerup and  Thorbjørnsen [169], where they considered only the case a, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A bit over a decade later,  Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [71] showed that the LUE moment recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) holds for all integers  k > −a − 1 with a > −1 a continuous real parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, recurrence  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) pertaining to the moments of the LOE and LSE can be seen to be a homogeneous  simplification of [71, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (43)], wherein the inhomogeneous terms depend on the moments  of the LUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Another simple observation is that for each of β = 1, 2, and 4, the m(L)  k  recurrences  displayed in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 involve one less term than the corresponding m(J)  k  recurrences  given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 — the terms in the recurrences are themselves drastically simpler, as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is in keeping with the loss of parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A similar simplification of terms can  be observed in the moment recurrences in the Gaussian case, again to be understood as a  consequence of the hierarchy implied by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we do not see a reduction in  the order of the moment recurrences when moving from the Laguerre to Gaussian ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, the spectral (and generalised) moments of the GUE satisfy a second order linear  recurrence first derived by Harer and Zagier in [174], while the spectral moments of the GOE  and GSE are characterised by fourth order linear recurrences given by Ledoux in [224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We  do not present them here, but note (as was mentioned earlier) that the strategies discussed  in this section recover the recurrences of [174], [224], with the added insight that they hold  for complex k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, our methods allow us to derive linear recurrence relations for  the β = 2/3 and β = 6 Gaussian ensembles’ spectral moments, which we now give.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2/3, 6 and k ∈ C such that Re(k) > 11, the moments of the Gaussian β  ensemble’s eigenvalue density satisfy the sixth order linear recurrence  6  ∑  l=0  d(G)  6,l  �κ − 1  4  �l  m(G)  k−2l = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  129  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  where  d(G)  6,0 = −4(k + 2),  d(G)  6,1 = 8(3k − 1)(3Nβ + 2),  d(G)  6,2 = 48(8 − 3k)Nβ(3Nβ + 4) + 49k3 − 216k2 + 92k + 320,  d(G)  6,3 = 4(k − 5)(3Nβ + 2)  �  24Nβ(3Nβ + 4) − 49k2 + 304k − 442  �  ,  d(G)  6,4 = 2(k − 5)3  �  294Nβ(3Nβ + 4) − 63k(k − 6) − 274  �  ,  d(G)  6,5 = 252(k − 5)5(3Nβ + 2),  d(G)  6,6 = 81(k − 5)7,  and we retain the definition Nβ = (κ − 1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Here, (x)n = x(x − 1) · · · (x − n + 1) is the falling  Pochhammer symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The initial terms m(G)  0  , m(G)  2  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m(G)  10  needed to determine the sequences  {m(G)  k  |β=2/3,6}k∈N are listed in [319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As we proceed to our discussion on 1-point recursions, let us conclude this subsection  with a couple of general remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The recurrences given above describe relations between  the complex-k generalised moments mk defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), so long as all moments  involved are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, the recurrences hold true when the mk are taken to  be (integer-k) spectral moments as defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), even though these spectral  moments fail to coincide with the generalised moments when considering the Gaussian,  symmetric Jacobi, or symmetric Cauchy ensembles, with k an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In these cases, the  definitions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) disagree only in that, according to the first definition, mk = 0  when k is odd — the relevant recurrences are then seen to hold trivially for odd integers k  due to their homogeneous structure and the fact that they run over every second integer-k  moment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) involves m(Cy)  k+6 , m(Cy)  k+4 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m(Cy)  k−4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The methods employed in [224] manifest a coupling between the GOE and GUE moments,  which does not arise naturally from our viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise for the coupling between the  LOE and LUE moments implied by the recurrence [71, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (43)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As made clear in [71],  both couplings can be traced back to a structural formula expressing the β = 1 eigenvalue  densities in terms of their β = 2 counterparts plus what can be regarded as rank one  corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This inter-relation was discussed briefly in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, where we saw that the context  underlying it is that the skew-orthogonal polynomials pertaining to the GOE and LOE can  130  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  be constructed from the Hermite and Laguerre orthogonal polynomials used to study the  GUE and LUE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' At present, there is no evidence implying a relation connecting  the moments of the β = 6 Gaussian ensemble to the moments of the GOE and/or GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  1-point recursions for the moment expansion coefficients  Following on from the preceding subsection, it is a simple matter to obtain 1-point recursions  of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) for the coefficients of the moment expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact,  all one needs to do is substitute the moment expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) into the recurrence  relations of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, now considering k large enough integers such that all involved moments  are well-defined according to definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), and then equate terms of equal order in  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There is however another (instructive) avenue towards the sought 1-point recursions  beginning at the differential-difference equations of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1: In the large N limit, the spectral  moments m(G)  2k , m(L)  k , and m(J)  k  do not converge, so the moment expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  may only be understood in a formal sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rectifying this issue by introducing the scalings  ˜m(G)  2k  := κ−kN−k−1 m(G)  2k ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  ˜m(L)  k  := κ−kN−k−1 m(L)  k ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  ˜m(J)  k  := m(J)  k /N,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  as is compliant with Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 (that is, ˜mk = �  R λk ˜ρ(λ) dλ), we observe that the (scaled)  resolvents  ˜W(G)  1  (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5),  ˜W(L)  1  (x) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), and W(J)  1 (x) act as generating functions for  { ˜m(G)  2k }k∈N, { ˜m(L)  k }k∈N, and { ˜m(J)  k }k∈N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, the scaled moments have  convergent large N expansions (finite sums in the Gaussian and Laguerre cases) of the  form ˜mk = ∑∞  l=0 ˜Mk,l N−l (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), with the expansion coefficients ˜Mk,l of the scaled moments  themselves generated by the resolvent expansion coefficients Wl  1(x) defined implicitly in  equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, substituting the large x expansions  W(G),l  1  (x) = 2l+1κl/2  ∞  ∑  k=0  ˜M(G)  k,l x−k−1,  W(L),l  1  (x) = κl/2  ∞  ∑  k=0  ˜M(L)  k,l x−k−1,  W(J),l  1  (x) =  ∞  ∑  k=0  ˜M(J)  k,l x−k−1  131  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  into the differential-difference equations of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and then comparing terms of like order in  x yields 1-point recursions on the ˜Mk,l (recall the diagram displayed at the end of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To obtain equivalent recursions for the unscaled moment expansion coefficients Mk,l, it is  enough to note from equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) that  M(G)  k,l = κk ˜M(G)  2k,l,  M(L)  k,l = κk ˜M(L)  k,l ,  M(J)  k,l = ˜M(J)  k,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Confirming agreement between the two derivations outlined above, flowing separately  from the results of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, respectively, serves as a consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' That aside,  the upcoming 1-point recursions are themselves motivated by the fact that they enable  efficient computation of the expansion coefficients Mk,l, assuming knowledge of sufficiently  many of them for small k, l ∈ N — we do not provide the necessary initial conditions here,  but note that they can be computed through the strategy discussed at the beginning of  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Compared to the recurrences of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, which run over the spectral moments mk, the  1-point recursions are faster at isolating the leading order behaviour of the mk, in the large  N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, changing viewpoint to that of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, the 1-point recursions  are interesting because they relate enumerations of ribbon graphs in a manner that has no  obvious combinatorial interpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we make only brief comments on the relevant ribbon  graphs here, deferring full discussion to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As mentioned earlier, our 1-point recursions agree with that of Harer and Zagier (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  in the GUE case and of Ledoux [224] in the GOE and GSE cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us recall that the  motivation for studying the former came from the interpretation of 2kM(GUE)  k,l  as the number  of ways of gluing the edges of a 2k-gon to form a compact orientable surface of genus l/2  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), while 4kM(GOE)  k,l  is known [165], [166], [251], [201] to be equal to the  number of such gluings that form a compact locally orientable surface of Euler characteristic  2 − l (see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' M(GSE)  k,l  has been given equivalent interpretations in [251], [57], which  can be understood through the β ↔ 4/β duality of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do not recount the  GOE, GUE, and GSE 1-point recursions here, for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, we do not report on  the symmetric Cauchy ensembles, nor the (symmetric shifted) Jacobi ensemble when β ̸= 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  though these have not been given in the literature before, they are relatively cumbersome to  display while being as equally easy to derive from the recurrences of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 as the recursions  given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our first result is hence on the Gaussian β ensembles with β = 2/3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  132  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Expand the spectral moments of the Gaussian ensemble according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  for β = 2/3, 6 and k ⩾ 6,  (k + 1)M(G)  k,l =  6  ∑  i=1  i  ∑  j=0  (κ − 1)2i−j  2i  fi,jM(G)  k−i,l−j,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  where  f1,0 = 3(6k − 1),  f1,1 = 2(6k − 1),  f2,0 = 36(4 − 3k),  f2,1 = 48(4 − 3k),  f2,2 = 49k3 − 108k2 + 23k + 40,  f3,0 = 108(2k − 5),  f3,1 = 216(2k − 5),  f3,2 = 3(5 − 2k)(98k2 − 304k + 189),  f3,3 = 2(5 − 2k)(98k2 − 304k + 221),  f4,2 = 441  2 (2k − 5)3,  f4,3 = 294(2k − 5)3,  f4,4 = 1  2(2k − 5)3 (126k(3 − k) − 137) ,  f5,4 = 189  2 (2k − 5)5,  f5,5 = 63(2k − 5)5,  f6,6 = 81  8 (2k − 5)7,  and all other fi,j are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We also set M(G)  k,l = 0 if l < 0 or l > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) is used to compute M(G)  k,0 , it reduces to  16(k + 1)M(G)  k,0 = 12(6k − 1)(κ − 1)2M(G)  k−1,0 − 36(3k − 4)(κ − 1)4M(G)  k−2,0  + 27(2k − 5)(κ − 1)6M(G)  k−3,0,  κ = 1/3 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  This is in keeping with the limiting scaled eigenvalue density ρ(G),0(λ) of the Gaussian  ensembles equalling the Wigner semi-circle law specified by the density  √  2 − λ2/π (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  supported on |λ| <  √  2: up to a scale factor the Catalan numbers are the even moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) can essentially be seen to be a four term linear recurrence for the  Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To the best of the author’s knowledge, it does not have a combinatorial  interpretation comparable to what is known for the traditional two term recurrence given in  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we note that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) and its three term analogue  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) are implied by repeated applications of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) with appropriate scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To expand the Laguerre and Jacobi ensembles’ spectral moments in N, we need to decide  how the parameters a and b scale with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We were faced with this same decision in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1,  where we opted to set a = ˆaκN and b = ˆbκN with ˆa, ˆb constant in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We continue with  this choice for consistency, with one caveat: Since the 1-point recursion in the LUE case  is relatively simple, we consider the slightly more general parametrisation a = ˆaκN + δa,  133  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  where ˆa, δa = O(1) — this gives us an opportunity to demonstrate the insight gained, at  the cost of complexity, from implementing this generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting this generalised  parametrisation together with the expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) into recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) yields the LUE  1-point recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 2, a = ˆaN + δa with ˆa, δa constant in N, and k ⩾ 2,  (k + 1)M(LUE)  k,l  = (2k − 1)  �  (2 + ˆa)M(LUE)  k−1,l + δaM(LUE)  k−1,l−1  �  − ˆa2(k − 2)M(LUE)  k−2,l + (k − 2)  �(k − 1)2 − δ2  a  �  M(LUE)  k−2,l−2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  where we set M(LUE)  k,l  = 0 if l < 0 or l > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When this is used to compute M(LUE)  k,0  in the case a = δa = O(1) and thus ˆa = 0, one  recovers the familiar Catalan recursion  (k + 1)M(LUE)  k,0  = 2(2k − 1)M(LUE)  k−1,0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  This recurrence is well known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [290]) and can be seen to be consistent with  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2: When a = O(1), we have from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='88) that γ1 = 1 so that the  Marˇcenko–Pastur density ρ(L),0(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) simplifies to √  4/λ − 1/(2π)χ0<λ<4, whose kth  spectral moment is exactly the kth Catalan number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Observe also that the Harer–Zagier  recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) reduces to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) upon setting l = 0 and noting that M(LUE)  k,0  = 2kM(GUE)  k,0  ,  this equality being implied by the fact that ρ(G),0(λ) = 2|λ| ρ(L),0(2λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In contrast to the four term recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23), equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) has a famous combina-  torial interpretation, which we now present in the language of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For k ∈ N and  β = 2, the expansion coefficients M(LUE)  k,0  count the number of unique planar ribbon graphs  that can be constructed from a 2k-gon1 [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To see that they are equal to the kth Catalan  number, observe that the set of planar ribbon graphs that can be constructed from a 2k-gon  are in bijection with the set of balanced parenthesisations involving k pairs of parentheses,  which are well known to be enumerated by the Catalan numbers [290];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we make the bijection  explicit by moving clockwise around the 2k-gon, starting at the top edge, assigning the label  1In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we will see that the ribbon graphs pertaining to the Laguerre ensembles must obey a certain  bicolouring constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This constraint is automatically satisfied when the ribbon graphs are planar, so we ignore  it for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  134  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  ‘1’ (open parenthesis) to the first free edge and ‘−1’ (corresponding closed parenthesis) to the  edge connected to it by a ribbon, ‘2’ and ‘−2’ to the next free edge and its ribbon-connected  counterpart, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As exemplified in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 below, the edges are then labelled — in  clockwise order starting at the top edge — by the sequence (i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i2k) with i1 = 1 and  j < l whenever 0 < ij < il or 0 < ij = −il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2: The first image relates a planar ribbon graph to the balanced parenthesisation  ()(()) by labelling the top edge ‘1’ and then, moving clockwise, labelling the tail of the  ribbon connected to the top edge ‘−1’, the head of the next ribbon ‘2’, the head of the final  ribbon ‘3’, and then the tails of these ribbons ‘−3’ and ‘−2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the second illustration, we  highlight the edges of the hexagon which fit the edge-marking criteria below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in addition to  the sixth edge in the clockwise ordering, which is special, an edge is eligible if it precedes  an edge that is the tail of some ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the final image, we show how to split the vertex  preceding the top edge in the clockwise ordering so that there are seven vertices available  for the vertex-marking procedure outlined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To elucidate the combinatorics underlying equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), we take the ribbon graphs  labelled in the manner described above and assign them markings in one of two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  first marking convention requires that one edge of the 2k-gon be marked, with eligible edges  being those labelled by ij with either j = 2k or j such that ij+1 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' that is, we mark either the  final edge in the clockwise ordering, or one that immediately precedes a negatively-labelled  edge (see the second illustration of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There are (k + 1) such eligible edges, so  that the set of k-ribbon graphs edge-marked in this fashion are enumerated by the left-hand  side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For the second marking convention, we split the vertex connecting  the first and last edges so that, in clockwise ordering, we have a vertex immediately preceding  135  1  3  2  3CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  the first edge, a vertex immediately following the last edge, and 2k − 1 vertices in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Labelling one of these 2k + 1 vertices by either of ±0 results in a set of 2(2k + 1)M(LUE)  k,0  vertex-marked ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since this number is just the right-hand side of equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) with k replaced by k + 1, it is unsurprising that there is a bijection between the set of  k-ribbon graphs edge-marked in the first described convention and the set of (k − 1)-ribbon  graphs vertex-marked in the second convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To map from the first set to the second,  delete the marked edge, the ribbon connected to it, and the edge at the other end of said  ribbon, so that each deleted edge collapses to a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Of these two edge-turned-vertices,  mark the one corresponding to the deleted edge that was positively-labelled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' label it ‘−0’  if the positively-labelled edge was marked, or ‘0’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, we split the vertex  connecting the first and last edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For the reverse mapping, split the marked vertex into  two and connect them with an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If the vertex was marked with a ‘−0’, insert a marked  edge immediately anticlockwise to the new edge, and connect the two with a ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If  the vertex was instead marked with a ‘0’, we need to travel clockwise to a second vertex,  turn it into a marked edge, and connect said edge to the first through a ribbon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' to uniquely  determine the second vertex, we simply pick the furthest one in the clockwise ordering  (stopping before reaching the first vertex and edge) that allows us to perform this procedure  whilst still producing a planar ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The planar ribbon graphs displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 below map to each other via  the bijection detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the left, we have a planar 3-ribbon graph labelled by the  sequence (1, −1, 2, 3, −3, −2), with a marking on the final edge ‘−2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the right, we have a  planar 2-ribbon graph labelled by the sequence (1, −1, 2, −2), with the vertex between edges  ‘−1’ and ‘2’ marked with a ‘+0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To map the left image to the right, we delete the marked  edge, together with its partner ‘2’ and the ribbon connecting them, then relabel the edges  3, −3 as 2, −2 and split the vertex connecting edges ‘1’ and ‘−2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since the original edge ‘2’  (we focus on the positively-labelled deleted edge) was nestled between edges ‘−1’ and ‘3’  and was unmarked, we mark the corresponding vertex with a ‘+0’ (if the edge marking in  the left image was at the edge ’2’, this vertex-marking would have been a ‘−0’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain the  left illustration from the right, we expand the vertex labelled ‘+0’ into an edge, the vertex d  into a marked edge, and connect the two with a ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, we join up the new marked  136  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  edge and the top edge ‘1’ and relabel the edges in the appropriate manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The new marked  edge has to be at vertex d because it is the furthest along in the clockwise ordering since  vertices a, b appear earlier than vertex ‘+0’ in said ordering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' note that picking vertex c would  result in a non-planar (genus one) ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3: Representatives of the sets of edge-marked planar 3-ribbon graphs with a  hexagon at their core and vertex-marked planar 2-ribbon graphs stemming from a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  These sets are in bijection and have cardinality given manifestly by, respectively, the left-  and right-hand sides of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) with k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When we remove the constraint l = 0, M(LUE)  k,l  retains a combinatorial interpretation if  we fix a = δa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, as discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, it enumerates ribbon graphs that are now  bicoloured and of genus l/2 [82] (when a = 0, M(LUE)  k,l  vanishes for all odd l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equivalently,  M(LUE)  k,l  |a=0 counts the number of ways of forming a compact orientable genus l/2 surface  by gluing together the edges of a 2k-gon whose vertices are alternately coloured black and  red, with the gluing respecting this bicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When a = 0, the general-l recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  reduces to a recursion that runs over even l, which was recently derived in [86, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1]  as a means of enumerating the aforementioned gluings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Keeping δa = 0, but letting ˆa ⩾ 0  be free, M(LUE)  k,l  |a=ˆaN counts the same ribbon graphs, with the red vertices now weighted  by (ˆa + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, M(LUE)  k,l  |a=ˆaN is a polynomial in (ˆa + 1) with the coefficient of (ˆa + 1)p  equalling the number of genus l/2 bicoloured gluings of the aforementioned 2k-gon that have  p distinct red vertices surviving the gluing procedure (again, we refer to §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In this a = ˆaN setting, recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) continues to run over even l so that m(LUE)  k  |a=ˆaN is  manifestly an odd function in N, in keeping with the second point of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  137  1  a  2  1  b  a  3  2  2  1  3  2  0  C  +0CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  A curious point to note is that even though the moment expansion coefficients M(LUE)  k,l  have combinatorial interpretations for k, l ∈ N and a = ˆaN with ˆa ⩾ 0, the recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  relating them does not presently have such an interpretation (at least not one comparable to  the bijection demonstrated in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, it may be the case that this 1-point  recursion describes a relation between the M(LUE)  k,l  |a=ˆaN which simply does not correspond  to a topological procedure at the level of the ribbon graphs themselves — ascertaining the  validity of this intriguing proposition remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar to the LUE case,  the moment expansion coefficients Mk,l of the GUE, GOE, GSE, LOE, and LSE also have  combinatorial meaning, to be detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, but the 1-point recursions characterising  them are yet to be understood in a combinatorial sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With the GUE, GOE, and GSE 1-point  recursions known from the works [174], [224], we now supply the analogous recursions for  the LOE and LSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For β = 1 and 4, a = ˆaκN with ˆa constant in N, and k ⩾ 4,  (k + 1)M(L)  k,l =  4  ∑  i=1  i  ∑  j=0  (κ − 1)jgi,jM(L)  k−i,l−j,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  where  g1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0 = (4k − 1) (ˆaκ + 4(κ − 1)) (κ − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0 = (3 − 2k)  �ˆa2κ2 + 2(ˆaκ + 4(κ − 1))2(κ − 1)2�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 = 2(2k − 3)ˆaκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 = (k − 1)  �  5k2 − 11k + 4  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0 = (4k − 11)(ˆaκ + 4(κ − 1))ˆa2κ2(κ − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 = 2(11 − 4k)(ˆaκ + 4(κ − 1))ˆaκ(κ − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 = 2(3 − k)  �  5k2 − 19k + 16  � (ˆaκ + 4(κ − 1))(κ − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0 = (4 − k)ˆa4κ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 = 4(k − 4)ˆa3κ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 = (k − 4)  �  5k2 − 32k + 47  � ˆa2κ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 = 2(4 − k)(k − 3)(5k − 17)ˆaκ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  g4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 = 4(1 − k) [(k − 4)(k − 3)]2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  and all other gi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='j are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We also set M(L)  k,l = 0 if l < 0 or l > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When a = 0, M(LOE)  k,l  is shown in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to count the same bicoloured ribbon graphs  as M(LUE)  k,l  except that the ribbons are now allowed to have M¨obius half-twists so that the  ribbon graphs are no longer guaranteed to be globally orientable, though they still have  Euler genus l (recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and see Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19 for examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in  138  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  keeping with the discussion comparing equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), we see that the three  term analogue of these equations obtained by setting a = l = 0 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6,  (k + 1)M(L)  k,0 = 4(4k − 1)(κ − 1)2M(L)  k−1,0 − 32(2k − 3)(κ − 1)4M(L)  k−2,0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  is, up to scaling, satisfied by the Catalan numbers when κ = 1/2 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, this recursion  can be used to verify that 2kM(LOE)  k,0  |a=0, equivalently 2−kM(LSE)  k,0  |a=0, is equal to the kth  Catalan number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, we observe that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) is equivalent to that obtained  by setting l = 0 in the GOE 1-point recursion given in [224, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now returning to the general-l case, let us compare the moment expansion coefficients  M(L)  k,l and M(G)  k,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We immediately observe that the former is somehow more complicated than  the latter: M(L)  k,l enumerates the same ribbon graphs as M(G)  k,l  but with the extra bicolouring  constraint described earlier and in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, so that M(L)  k,l |a=0 ⩽ M(G)  k,l and when a = ˆaκN ̸= 0,  (ˆa + 1) acts as a weight that keeps track of the ratio of red to black vertices in the ribbon  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note also that the presence of the parameter a results in the expansion coefficients  M(L)  k,l and the 1-point recursions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) being more complex than their Gaussian  counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This increase in complexity is linked to the fact that the Laguerre ensembles  sit above the Gaussian ensembles in the hierarchy implied by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We posit that  since the Jacobi ensembles inhabit the tier immediately above the Laguerre ensembles in  the aforementioned hierarchy, it is not unreasonable to expect that the M(J)  k,l (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) count  some type of ribbon graphs that include, as special cases, the ribbon graphs enumerated by  the M(G)  k,l  or M(L)  k,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, there is no such construction currently known in the literature,  with attempts at replicating the constructions of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 thwarted by the fact that the  Jacobi matrices of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 involve inverse matrices whose entries are not compatible  with the Isserlis–Wick theorem (see the discussion following Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It  is nonetheless the author’s belief, which we justify below (see also the discussion in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  regarding the works [326], [25], [73], [159], [160]), that the M(J)  k,l should have combinatorial  interpretations in line with what is known for the Gaussian and Laguerre ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To motivate the study of the Jacobi ensembles’ moment expansion coefficients M(J)  k,l with  a combinatorial flavour, let us review some properties of the classical matrix ensembles that  are relevant in the combinatorial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To begin, recall from the paragraph preceding  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 that m(J)  k , consequently M(J)  k,l , reduces to m(L)  k , respectively M(L)  k,l , when taking  139  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  b → ∞ (possibly by setting b = ˆbκN and working in the large N limit), so that if M(J)  k,l  were to enumerate a class of ribbon graphs, these ribbon graphs would degenerate to those  enumerated by M(L)  k,l in the large b limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is understood to be a simple consequence of  the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) presented in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, which is key to comparing the Laguerre and Jacobi  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The other limit given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, relating the Jacobi and Gaussian ensembles,  has a cleaner form when written in terms of the symmetric shifted Jacobi ensemble:  lim  a→∞ w(sJ)(λ/√  a)  ��  a=b = w(G)(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A similar observation can be made by comparing the identities  ρ(G),0(λ) =  √  2(1 − λ2/2)ρ(sJ),0(λ/  √  2)  ��  a=b=0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  ρ(G),0(λ) = |λ|(1 − λ2/2)ρ(J),0(λ2/2)  ��  a=b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  Recalling the discussion following equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) is reminiscent of the  analogous relation between ρ(G),0(λ) and ρ(L),0(λ), suggesting that while the GUE can be  compared at leading order to the sJUE by a linear change of variables, the LUE is more  closely related to the JUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If we expand µ(sJ)  k  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) as  µ(sJ)  k  =  ∞  ∑  l=0  ∆M(sJ)  k,l N1−l  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  and define ∆M(J)  k,l := M(J)  k,l − M(J)  k+1,l, equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) imply that  2k∆M(J)  k,0  ��  a=b=0 = −2k+1∆M(sJ)  k,0  ��  a=0 = M(G)  k,0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  which links the differences of the moments of ρ(J),0(λ)|a=b=0 and ρ(sJ),0(λ)|a=b=0 to the  Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Combining these observations together, it seems that there is merit in  studying the moment expansion coefficients of both the un-shifted and symmetric shifted  Jacobi ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' At this point in time, a good approach for progressing this line of study  is to find more ways of comparing the moment expansion coefficients Mk,l of the classical  matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To this end, we now derive the 1-point recursions satisfied by M(JUE)  k,l  ,  ∆M(JUE)  k,l  , and ∆M(sJUE)  k,l  , fixing β = 2 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Fix β = 2, a = ˆaN, and b = ˆbN, with ˆa, ˆb = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for k ⩾ 3,  k(ˆa + ˆb + 2)2M(JUE)  k,l  = k(k − 1)2M(JUE)  k,l−2 +  3  ∑  i=1  �  hi,0M(JUE)  k−i,l + hi,1M(JUE)  k−i,l−2  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  140  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  where  h1,0 = 2(4k − 3)(ˆa + ˆb + 1) +  �  3ˆa(k − 1) + ˆbk  �  (ˆa + ˆb),  h1,1 = (1 − k)(3k2 − 8k + 6),  h2,0 = 3ˆa2(2 − k) + (3 − 2k)  �  (ˆa + 2)(ˆb + 2) − 2  �  ,  h2,1 = (k − 2)(3k2 − 10k + 9),  h3,0 = ˆa2(k − 3),  h3,1 = (3 − k)(k − 2)2,  and we set M(JUE)  k,l  = 0 if l < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The derivation of this recursion is slightly different to that described earlier, so we  supply the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substitute expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) into the recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), along with the  substitutions a = ˆaN and b = ˆbN, and then equate terms of order 3 − l in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that  with our choice of a and b, the coefficients d(J)  2,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , d(J)  2,3 all contain a term of order two and  a term of order one in N, and no other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The O(N2) terms of d(J)  2,i correspond to the  coefficients hi,0 of M(JUE)  k−i,l  in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), while the terms of order one in N correspond  to the coefficients hi,1 of M(JUE)  k−i,l−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 leads directly to the 1-point recursion on the ∆M(JUE)  k,l  |a=ˆaN, b=ˆbN =  M(JUE)  k,l  |a=ˆaN, b=ˆbN − M(JUE)  k+1,l |a=ˆaN, b=ˆbN, while applying the inverse of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) yields  the equivalent relation on the ∆M(sJUE)  k,l  |a=ˆaN (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Alternatively, the latter can be derived  from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) by following the prescription at the beginning of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the setting of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 with ∆M(JUE)  k,l  = M(JUE)  k,l  − M(JUE)  k+1,l and ∆M(sJUE)  k,l  defined implicitly by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30), we have for k ⩾ 2 the 1-point recursions  (k + 1)(ˆa + ˆb + 2)2∆M(JUE)  k,l  = (2k − 1)  �  ˆa(ˆa + ˆb + 2) + 2(ˆb + 1)  �  ∆M(JUE)  k−1,l  + (2 − k)ˆa2∆M(JUE)  k−2,l + k2(k + 1)∆M(JUE)  k,l−2  − k(k − 1)(2k − 1)∆M(JUE)  k−1,l−2 + (k − 2)(k − 1)2∆M(JUE)  k−2,l−2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  and  4(k + 1)(ˆa + 1)2∆M(sJUE)  k,l  = 2(2k − 1)(2ˆa + 1)∆M(sJUE)  k−1,l  + (k + 1)(2k + 1)2∆M(sJUE)  k,l−2  + 4k2(2k − 1)∆M(sJUE)  k−1,l−2 + (k − 1)(2k − 1)(2k − 3)∆M(sJUE)  k−2,l−2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  with ∆M(JUE)  k,l  , ∆M(sJUE)  k,l  set to zero for all l < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  141  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  There are three immediate observations relating to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Firstly, the recursions therein run over even l, similar to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, when  the first few moments m(JUE)  k  , m(sJUE)  k  with k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 3 (known from [96], [242], [142]) are  expanded as series in 1/N, they do not contain terms of even powers in N, so the recursions  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) are trivially satisfied when l is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As with Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5, this feature  is due to our choice of taking β = 2 and a, b ∝ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our second observation is that, unlike  the analogous 1-point recursions for the GUE and LUE moment expansion coefficients,  the recursions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 show that M(JUE)  k,l  , ∆M(JUE)  k,l  , ∆M(sJUE)  k,l  depend, respectively, on M(JUE)  k,l−2 , ∆M(JUE)  k,l−2 , ∆M(sJUE)  k,l−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, in contrast to the Gaussian and  Laguerre cases, knowledge of m(JUE)  k′  , ∆m(JUE)  k′  , µ(sJUE)  k′  with k′ < k is not enough to determine  M(JUE)  k,l  , ∆M(JUE)  k,l  , ∆M(sJUE)  k,l  for l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our final observation is that of the three recursions  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34), the final one is the simplest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  If we are to assume that the moment expansion coefficients of the Jacobi ensembles  count some type of surface similar to those counted by the GUE and LUE spectral moments,  comparing the corresponding 1-point recursions may shed light on what these surfaces  actually are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, the fact that the recursions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  run over even l suggests that l/2 might play the role of genus in the Jacobi case, as it does in  the Gaussian and Laguerre cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, the second observation given above  could give a hint as to what sets the JUE apart from the GUE and LUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since setting a = 0  in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 yields a simpler recursion that retains an interpretation in terms of ribbon  graphs, we set a = b = 0 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 for comparison:  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the context of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, setting a = b = 0 reduces  recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) to  4kM(JUE)  k,l  = k(k − 1)2M(JUE)  k,l−2 + 2(4k − 3)M(JUE)  k−1,l  + (1 − k)(3k2 − 8k + 6)M(JUE)  k−1,l−2 + 2(3 − 2k)M(JUE)  k−2,l  + (k − 2)(3k2 − 10k + 9)M(JUE)  k−2,l−2 + (3 − k)(k − 2)2M(JUE)  k−3,l−2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) to  4(k + 1)∆M(JUE)  k,l  = 2(2k − 1)∆M(JUE)  k−1,l + k2(k + 1)∆M(JUE)  k,l−2  − k(k − 1)(2k − 1)∆M(JUE)  k−1,l−2 + (k − 2)(k − 1)2∆M(JUE)  k−2,l−2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  142  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recurrence Relations for the Moments of the Classical Matrix Ensembles  and recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) to  4(k + 1)∆M(sJUE)  k,l  = 2(2k − 1)∆M(sJUE)  k−1,l  + (k + 1)(2k + 1)2∆M(sJUE)  k,l−2  + 4k2(2k − 1)∆M(sJUE)  k−1,l−2 + (k − 1)(2k − 1)(2k − 3)∆M(sJUE)  k−2,l−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  Comparing these forms to the GUE analogue (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) and the result [86, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1] of  setting a = 0 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5,  (k + 1)M(LUE)  k,l  = 2(2k − 1)M(LUE)  k−1,l + (k − 2)(k − 1)2M(LUE)  k−2,l−2,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  we note that equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) contain familiar terms, whereas the same is not  true for equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Setting equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) aside, we see that upon ignoring the terms  with indices (k, l − 2) and (k − 1, l − 2), equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) has the same structure as equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38), while equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) is almost equivalent to the Harer–Zagier recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Hence, it seems that the JUE should be compared to the LUE and the sJUE should be com-  pared to the GUE (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the discussion following equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In addition,  the coefficients of the ignored terms contain, respectively, the factors (k + 1) and (2k − 1),  which suggests that we should group them with the terms with indices (k, l) and (k − 1, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As expected, setting l = 0 in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) shows that 2k∆M(JUE)  k,0  |a=b=0  and −2k+1∆M(sJUE)  k,0  |a=0 satisfy the two term Catalan recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25), which is in keeping  with equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moving past these observations, the challenge now is to extend the  combinatorial interpretation of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) to the general-l recursions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned earlier, this remains an open problem that we are unable  to address here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless, some comments on promising directions are given in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As  we move on to the next section, let us make a final remark: Experimentation for 0 ⩽ k ⩽ 10,  0 ⩽ l ⩽ 5 suggests that for each k, l ⩾ 0, −22(k+l)+1∆M(sJUE)  k,l  |a=0 is a positive integer, which  supports the idea that these scaled differences of moments are equal to the cardinality of  some class of combinatorial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, it can be seen that these positive integers  grow extremely quickly as k, l are increased, relative to what is seen for their GUE and LUE  counterparts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' to be more concrete, comparing equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) shows that for  all k, l ⩾ 0,  −22(k+l)+1∆M(sJUE)  k,l  |a=0 ⩾ M(GUE)  k,l  ⩾ M(LUE)  k,l  |a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  143  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Established Results on the Moments of the Classical Matrix  Ensembles  In the previous section, we derived recursions characterising the spectral moments of the  classical matrix ensembles, along with their complex-k generalisations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) and their  expansion coefficients Mk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To summarise Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, our methodology consisted  of using the theory of Selberg correlation integrals to obtain differential equations satisfied  by the eigenvalue densities ρ(x) and resolvents W1(x) of the classical matrix ensembles —  applying integration by parts on the former set of differential equations or inserting the  expansion W1(x) = ∑∞  k=0 mk/xk+1 in the latter set resulted in the sought recursions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Some  of the recursions derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 have already been given in the literature, but our  approach involving Selberg correlation integrals is distinct and is able to treat all of the  classical matrix ensembles in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this section, we give a select review on  recent studies that provide characterisations of the moments of the classical matrix ensembles,  focusing particularly on those that are different to the results of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 (classical results  on this topic are reviewed in a plethora of texts, including [238], [119]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Seeing as this  gives us an opportunity to showcase connections to various fields of mathematics, we  compartmentalise our review by methodology rather than outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Results from skew-orthogonal polynomial theory  A straightforward approach to computing the spectral moments is to simply integrate  monomials against closed form expressions for the corresponding eigenvalue densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  This approach was taken by Livan and Vivo in 2011 [229], with their starting point being  expressions for the classical Laguerre and Jacobi ensembles’ eigenvalue densities known  from (skew-)orthogonal polynomial theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling the discussion of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, the idea then  is to take ρ(λ) to be given by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='67) with x = y = λ when β = 2, and by this  plus a correction term when β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Integrating these expressions against λk for k ∈ N,  Livan and Vivo gave exact expressions for the moments of the classical Laguerre and Jacobi  ensembles in terms of sums of ratios of gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' At this same time, in a parallel  but independent work, Mezzadri and Simm [240] derived equivalent expressions for the  144  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Established Results on the Moments of the Classical Matrix Ensembles  spectral moments mk of the classical Gaussian, Laguerre, and Jacobi ensembles, moreover  showing that their expressions hold true for all k ∈ C such that the mk are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In order to ease the required integrations against the aforementioned expressions for the  eigenvalue densities ρ(w)(λ), a key idea of Mezzadri and Simm was to use polynomials that  are orthogonal with respect to the perturbations λkw(λ) of the classical weights w(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The moment recurrences given in [169], [223], [224], [71] were also obtained through  consideration of (skew-)orthogonal polynomial theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the 2003 work [169], Haagerup and  Thorbjørnsen used elementary identities satisfied by the Hermite and Laguerre orthogonal  polynomials to express the exponential moment generating function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18))  R1(x) :=  �  supp ρ exp(xλ)ρ(λ) dλ,  for the GUE and LUE cases in terms of (confluent) hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They then  transformed known differential equations satisfied by these hypergeometric functions into  differential equations characterising R1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From there, Haagerup and Thorbjørnsen obtained  recurrences for the GUE and LUE (positive-integer) spectral moments through essentially  the same strategy as used in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, thereby recovering the recursion of Harer and Zagier  [174], and finding its LUE analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Ledoux [223] extended these ideas to the JUE case  a year later, and to the GOE case [224] in 2009 — clever manipulations were needed to  handle the correction term that must be added to the GUE eigenvalue density to obtain  its GOE analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In 2016, Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [71] extended the work of Ledoux to derive an  inhomogeneous moment recurrence for the LOE, where the inhomogeneous terms were given  in terms of the LUE moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This recurrence, together with the LUE moment recurrence  given in [169], was shown to hold for negative-integer moments (m−k with 0 < k < a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Results from symmetric function theory  Of the classical β ensembles (recall §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), only the β = 1, 2, and 4 regimes can be studied  through (skew-)orthogonal polynomial theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To compute the spectral moments in the  general β case, other approaches must be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One option is to use loop equation analysis  to iteratively compute the resolvent expansion coefficients W0  1(x), W1  1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) and  then interpret them as generating functions for the moment expansion coefficients Mk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the Gaussian and Laguerre cases, the spectral moments are polynomials in N, so this  145  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  procedure can be used to compute the spectral moment mk for any given k, assuming enough  computation power is available;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in the case of the Jacobi and Cauchy β ensembles, the best  one can hope for is a truncation of the large N expansion of the spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do not  review the loop equation formalism any further here since it will be revisited in Chapter 4,  but note that the Gaussian, Laguerre, and Jacobi β ensembles were studied in this manner in  the 2014 and 2017 works [319], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  While the loop equation formalism can be used to compute (the leading order behaviour  of) spectral moments of the classical β ensembles, it does not provide any closed form  expressions for these moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Such closed form expressions can however be obtained in  the general β setting through symmetric function theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The relevance of said theory should  not be surprising since equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) shows that the kth spectral moment is given by the  integral of the symmetric polynomial ∑N  i=1 λk  i multiplied by the appropriate eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Some well known bases for the ring of symmetric polynomials are the sets of monomial  symmetric polynomials, Schur polynomials, zonal polynomials, and the Jack polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As an algebra, it is also generated by the sets of power-sum symmetric polynomials and  elementary symmetric polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We refer the reader to [232] for definitions of these  polynomials, but note that the elementary symmetric polynomials were mentioned earlier in  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Of particular interest to us is the set of Jack polynomials of type ��C”, which differ  from those of type “J” and “P” by a choice of normalisation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [92]) and are written as  C(α)  σ (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  They are indexed by partitions σ of integers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , σk) with k, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , σk ∈ N  and σ1 ⩾ σ2 ⩾ · · · ⩾ σk > 0) and a positive real parameter α not to be confused with that  associated with the Cauchy ensemble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' setting α = 1 recovers the Schur polynomials while  α = 2 corresponds to the zonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the 1997 work on Selberg correlation integrals [199], Kadell gave an explicit closed  form expression, in terms of ratios of gamma functions, for the integral  �  [0,1]N C(2/β)  σ  (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) p(J)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) dλ1 · · · dλN,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  where p(J)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) is the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) of the Jacobi β ensemble and  σ is any given partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the same year, Baker and Forrester [27] showed that when  146  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Established Results on the Moments of the Classical Matrix Ensembles  w(λ) is either the Gaussian or Laguerre weight (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9), the above integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) with p(J)  replaced by p(w) is equal to the generalised Hermite (up to a minus sign), respectively  Laguerre, polynomial of index σ evaluated at zero, these generalised polynomials being the  multivariable symmetric polynomials that are orthogonal with respect to the inner products  ⟨ f, g⟩(w) :=  �  RN f (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN)g(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) p(w)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) dλ1 · · · dλN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the Jacobi case, one can surmise from [27] a result analogous to that of Kadell [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Around 2003 [92], Dumitriu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' used the results of [199], [27] (among others) to devise  algorithms, implemented in the MOPS package [96], for computing the spectral moments  of the Gaussian, Laguerre, and Jacobi β ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These algorithms essentially boil down  to expressing the polynomial ∑N  i=1 λk  i as a linear combination of Jack “C” polynomials so  that the problem of computing the spectral moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) reduces to the computation  of integrals of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One should note that these algorithms are not equivalent  to closed form expressions for the spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, in the 2017 work [242],  Mezzadri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' provide such closed form expressions in the Jacobi and Laguerre cases by  way of giving explicit formulae for the coefficients in the aforementioned linear combination  of Jack “C” polynomials that is equal to ∑N  i=1 λk  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, they express m(J)  k  and m(L)  k  as  sums over partitions of k with each summand being an explicit ratio of gamma functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Prompted by Fyodorov and Le Doussal, they also show how to use a relation between  �  C(α)  σ (1/λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 1/λN)  �  σ and  �  C(α)  σ′ (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN)  �  σ′ to extend their expressions for m(J)  k , m(L)  k  to the case that k is a negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, these expressions for the integer moments of  the Jacobi and Laguerre β ensembles as sums of ratios of gamma functions were given by  Fyodorov and Le Doussal in the independent work [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, this latter work contains  contour integral representations for m(J)  k , m(L)  k  (k ∈ Z sufficiently large) due to Borodin and  Gorin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Before moving on, let us finally mention that Novaes also studied the negative-integer  moments of the LUE using Schur polynomials in the 2015 work [267].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Results in terms of hypergeometric orthogonal polynomials  In the recent work [72], Cunden at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' made the novel observation that in the classical  Gaussian, Laguerre, and Jacobi cases, simple linear combinations of the complex-k moments  mk (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), interpreted as functions of k and renormalised by factors dependent on N, are  147  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  hypergeometric orthogonal polynomials in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To be precise, they showed that when β = 2 and  N is fixed, m(G)  k  is essentially a Meixner–Pollaczek polynomial in k, m(L)  k  is a continuous dual  Hahn polynomial, and ∆m(J)  k  = m(J)  k  − m(J)  k+1 is a Wilson polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These three polynomials  belong to the Askey scheme of hypergeometric orthogonal polynomials, with the first two  being degenerations of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' prove this observation in the Gaussian  and Laguerre cases by seeing agreement in the hypergeometric function representations  of the moments and the corresponding polynomials when k ∈ N, and then extending this  agreement to k ∈ C through an application of Carlson’s theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in the  Jacobi case, lacking such a representation in terms of hypergeometric functions, they derive  the analogue of recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) for ∆m(J)  k  and show that this recurrence, after appropriate  scaling, also characterises the Wilson polynomials in a unique sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' By noting that certain  linear combinations of the β = 1, 4 moments are known to be given by linear combinations  of their β = 2 counterparts, Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' further show within [72] how the polynomial  characterisations of the β = 2 moments extends to the linear combinations considered in the  β = 1, 4 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It was shown in [72] that the LUE moments and JUE moment differences satisfy  the following reciprocity laws:  m(L)  −k−1 =  �  k  ∏  j=−k  1  a − j  �  m(L)  k ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  ∆m(J)  −k−1 =  �  k  ∏  j=−k  a + b + 2N − j  a − j  �  ∆m(J)  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  It is not immediately obvious that there exist similar laws for the moments of the orthogonal  or symplectic ensembles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' one can experiment using data computable from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is believed that this is a consequence of three term recurrences, as seen  in the β = 2 cases, being simpler than their five term (β = 1, 4) analogues in an integrable  sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, this is the reason that Cunden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' were able to place the β = 2 moments  m(G)  k  , m(L)  k , m(J)  k  in the Askey scheme of hypergeometric orthogonal polynomials, but were  only able to do so for linear combinations of the moments when taking β = 1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Soon after the work [72], Assiotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' showed [23] that the complex-k sum of moments  µ(CyUE)  k  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) of the symmetric Cauchy unitary ensemble, again understood as a function of  148  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Established Results on the Moments of the Classical Matrix Ensembles  k with fixed N and properly renormalised, is also a hypergeometric orthogonal polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  This time, the relevant polynomials are the continuous Hahn polynomials, which are also  degenerations of the Wilson polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling that the hypergeometric functions are  defined by  pFq  �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ap  b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , bq  ���� z  �  :=  ∞  ∑  n=0  (a1)(n) · · · (ap)(n)  (b1)(n) · · · (bq)(n)  zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  with (x)(n) = x(x + 1) · · · (x + n − 1) denoting the rising Pochhammer symbol, the continuous  Hahn polynomials are defined as [207]  Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, b, c, d) := in (a + c)(n)(a + d)(n)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3F2  �−n, n + a + b + c + d − 1, a + ix  a + c, a + d  ���� 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  Assiotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' proceed by deriving a differential equation characterising the eigenvalue  density ρ(Cy)(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, 2) of the CyUE, using this differential equation to obtain recurrence  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) on the sums of moments µ(Cy)  k  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), and then confirming that this recurrence also  characterises the continuous Hahn polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, they show that  µ(CyUE)  k  =  Γ (k + 1/2) Γ (α − k − 1/2)  Γ (α + 3/2) Γ (−α − 1/2) √π  i1−N  2  Γ (1/2 − α − N) α(2α + N)  × SN−1  �  −i(k + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1, 1  2 + α, 1, 1  2 + α  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  with SN−1 defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We provide now the analogue of the above result of Assiotis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' in the case of the  symmetric shifted Jacobi unitary ensemble corresponding to the weight w(sJ)(λ)  ��  a=b (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us first recall from the third point of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that ρ(sJUE)(λ)  ��  a=b is equal to (1 − λ2)a  multiplied by a polynomial of order N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we see from equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  that  ˜µk := Γ(k + N + a + 3/2)  Γ(k + 1/2)  µ(sJUE)  k  ���  a=b  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  is a polynomial in k of order N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting this scaling into recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) results in  the simpler recurrence  (2k + 4)[(2k + 3) − 2(a + N)] ˜µk+1 + 2[(2k + 2)2 − 2N(N + 2a)] ˜µk  + (2k)(2(N + a + k) + 1) ˜µk−1 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  149  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  valid for k ∈ C with Re(k) > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now, let us constrain the continuous Hahn polynomials  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) by defining  sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, b) := Sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, b, a, b)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  and observe that this polynomial satisfies the recurrence [1, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15]  A(x)sn(x + i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, b) − [A(x) + C(x) − n(n + 2Re(a + b) − 1)] sn(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, b)  + C(x)sn(x − i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, b) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  with  A(x) = (x + ia)(x + ib),  C(x) = (x − ia)(x − ib).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For k ∈ C with Re(k) > −1/2 and ˜µk defined by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), we have  ˜µk =  ˜µ0i1−N  N(3/2 − (a + N))(N−1) sN−1  �  i(k + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1, 1  2 − (a + N)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  with  ˜µ0 = Γ(N + a + 3/2)  Γ(1/2)  �2N(a + N)(2a + N)  1 − 4(a + N)2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  Thus, the difference of moments µ(sJUE)  k  of the symmetric shifted Jacobi unitary ensemble is given in  terms of the continuous Hahn polynomials according to  µ(sJUE)  k  =  ˜µ0i1−NΓ(k + 1/2)  NΓ(k + N + a + 3/2)(3/2 − (a + N))(N−1) sN−1  �  i(k + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1, 1  2 − (a + N)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) is obtained by combining equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Comparing  equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) shows that  q(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, N) := CN,a sN−1  �  i(k + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1, 1  2 − (a + N)  �  ,  with CN,a independent of k, satisfies the recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain agreement with ˜µk at  k = 0, we set  CN,a =  ˜µ0  sN−1  �  i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1, 1  2 − (a + N)  � =  ˜µ0 i1−N  N(3/2 − (a + N))(N−1) ,  where the second equality follows from definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) of the continuous Hahn polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  With this choice, both q(k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' a, N) and ˜µk are polynomials of order N − 1 in k that coincide at  k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As they both satisfy recurrence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8), they also coincide at k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This agreement  is extended to k ∈ C via Carlson’s theorem (recall Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  follows from substituting equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  150  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  The Isserlis–Wick Theorem and Ribbon Graphs  A major motivation for studying ribbon graphs (recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) is their relation to  topological (hyper)maps, which we briefly discuss throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For now, we note  that the term ‘map’ is not synonymous with ‘function’ or ‘mapping’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, topological  maps are so-called because they are reminiscent of the geographic maps of countries that  one would find in an atlas — they are graphs embedded in surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These topological  maps have been studied by combinatorialists and geometers since the 1960s [302], [198],  but became prominent in algebraic geometry around 1984 due to Grothendieck’s work on  dessins d’enfants (children’s drawings) [284] — technically, a topological map is equivalent to  a ‘clean’ dessin (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 at the end of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The connection between ribbon graphs (equivalently, topological maps) and random  matrix theory is that, through consequences of the Isserlis–Wick theorem that we will soon  expound, random matrix theory provides a neat method for enumerating ribbon graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the  upshot here is that the pairwise polygon-edge identifications represented by the ribbons of a  ribbon graph can be interpreted as pairings of random matrix entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Algebraic geometers  and mathematical physicists have used this relationship in a number of intriguing ways, of  which we mention a select few:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let Ms  g denote the moduli space of genus g compact Riemann surfaces with s marked  points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the (6g − 6 + 2s)-dimensional space of parameters that determine the  complex structure of such surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using the uniqueness of the Jenkins–Strebel  quadratic differential on compact Riemann surfaces [193], [291], [292], Harer [173]  showed how to canonically assign a genus g, orientable, connected, metrised ribbon  graph built from s polygons to each point of Ms  g (here, metrised means that each  ribbon is labelled with a length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This had the consequence of endowing Ms  g with the  structure of a simplicial complex (each k-simplex corresponding to a k-ribbon graph  representative of all metrised ribbon graphs that reduce to it upon ‘forgetting’ ribbon  lengths), which enabled Harer and Zagier [174] to reduce the problem of computing  the (orbifold) Euler characteristic of Ms  g to that of enumerating ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  treatment was extended to the moduli space of real algebraic curves in [165], [166],  with the relevant ribbon graphs now allowed to be non-orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  151  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In string theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [274] for an introduction), one replaces particles with strings,  which are topologically lines or circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The trajectory of a string is called a worldsheet,  which is a surface with boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Considering only closed (circular) strings, a  worldsheet is a surface of, say, genus g with s punctures — the handles counted by  the genus arise from strings splitting and combining, while the punctures are simply  contractions of the boundaries that correspond to the initial and final states of strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In  the theory of 2D quantum gravity, one considers a generic worldsheet with its structure,  that is, a choice of metric, left undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The partition function is formulated as an  integral over the space of all possible worldsheet metrics, with the integrand given in  terms of aspects of the worldsheet that depend on the choice of metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This integral is  difficult to make sense of because the space of worldsheet metrics is unwieldy for many  reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In 1990, progress was made by the independent works [52], [89], [168], where  the idea was to discretise the worldsheet and approximate it by triangulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  topological maps with trivalent vertices or ribbon graphs built purely from triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When considering the limit of infinitely many triangles, these discrete approximations  improve to the point that each choice of triangulation corresponds to a worldsheet  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Consequently, the relevant partition function can essentially be written as a  sum over orientable, connected ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another approach to 2D quantum gravity, called topological gravity, has partition func-  tion equal to a generating function for certain intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These intersection  numbers are given by integrals over the compactifications M  s  g of the moduli spaces  defined in point 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A 1991 conjecture of Witten [322] is that topological gravity  should be equivalent to the formalism outlined in point 2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This conjecture was  famously proven by Kontsevich the following year [211] using the ideas of Harer that  we reviewed in point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In short, Kontsevich used the structure of Ms  g as a simplicial  complex (with each simplex corresponding to a specific ribbon graph) in order to relate  the aforementioned intersection numbers to explicit sums over ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For detailed surveys of these topics, see [83], [250], [220], [248].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The studies listed above share the philosophy of reducing a difficult problem to that  of enumerating ribbon graphs, and then using random matrix theory to perform said  152  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  enumerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In contrast to this philosophy, the literature also contains many examples  of using ribbon graphs (and similar objects) to diagrammatically encode calculations of  the (mixed) moments and cumulants defined through equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Naturally, such examples were first seen in the physics literature [304], [53], [287], [192],  since both random matrix theory and diagrammatic theories [179], [54] were prevalent in  the field during the mid-twentieth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, Wick’s theorem [312] on reducing  products of creation and annihilation operators to sums of products of creation-annihilation  operator pairings was well known to quantum field theorists and easily extends to treatment  of similar such products of random matrix entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, the required generalisation  pertaining to products of normally distributed random variables was actually known much  earlier to probability theorists as Isserlis’ theorem [186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us now state a version of this  theorem that is suitable for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 (Isserlis–Wick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For k ∈ N, let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xk be centred normal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, if k is  odd, the expectation ⟨x1 · · · xk⟩ is equal to zero, while for k even, it decomposes as  ⟨x1 · · · xk⟩ = ∑  σ∈Pk  �  xσ(1)xσ(2)  �  · ·  �  xσ(k−1)xσ(k)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  where Pk is the set of all permutations of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k} such that for all odd integers i, j with i < j,  σ(i) < σ(i + 1) and σ(i) < σ(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The key idea of this section, in conjunction with the Isserlis–Wick theorem, is that for  random matrices X that can be expressed as sums and products of Ginibre matrices (recall  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), Tr Xk is given by a sum of products of normally distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, recalling Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, the (mixed) moments and cumulants of the GOE and LOE can  be simplified using the Isserlis–Wick theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' graphically, the covariances ⟨xσ(j)xσ(j+1)⟩ in  the summand of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) are to be represented by ribbons connecting edges labelled  xσ(j) and xσ(j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' By linearity of the expectation, the Isserlis–Wick theorem also holds when  the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xk are drawn from mean-zero complex normal distributions, so it can  be used to study the GUE and LUE, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar reasoning applies in the GSE and LSE  cases, too [251], [57], though we do not discuss these cases for brevity (instead, we appeal to  the β ↔ 4/β duality of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Unfortunately, the Isserlis–Wick theorem cannot be used to relate ribbon graphs to any of  the other ensembles studied thus far in this thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, the Gaussian β ensembles with  153  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  β = 2/3, 6 do not have matrix realisations in terms of Ginibre matrices and must instead  be understood through the general-β constructions of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, there is no relation  between the Cauchy and Ginibre ensembles, while matrix realisations of the Jacobi ensembles  involve inverse Ginibre matrices (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, it should be  noted that the combinatorial approach of [326] was used in [25] to express m(JOE)  k  in terms  of the negative-integer moments of the LOE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the latter do not have any known ribbon graph  interpretations (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 for some further discussion on this point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we show how the Isserlis–Wick theorem leads to interpretations of the (mixed)  moments and cumulants of the GUE in terms of (globally) orientable ribbon graphs, and  then explain how the GOE analogues are similarly related to locally orientable ribbon  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we repeat these exercises for the LUE and LOE, detailing a necessary  bicolouring constraint on the vertices of the polygons that are present in the relevant  ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we combine the ideas of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to derive  ribbon graph representations for the mixed moments and cumulants of the Hermitised and  antisymmetrised matrix products introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, thereby justifying the existence  of the large N expansions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) for the corresponding connected n-point correlators Wn  (this latter point being significant in the context of Chapter 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We will be relaxed with our  referencing since our discussion will be an amalgamation of new results, results already  cited, and results that are most likely known to the community but unpublished in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  That said, let us note that the exposition of the first half of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 is equivalent to that of  [36], [329], while the first half of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 contains statements that loosely follow from [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The second half of both of these subsections discuss concepts that have been studied and  reviewed in [57], [219] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The contents of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 are original and based  on the ongoing work [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Moments of the Gaussian unitary and orthogonal ensembles  Let us now demonstrate how the Isserlis–Wick theorem can be used to express the spectral  moments m(GUE)  k  as sums over certain (k/2)-ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Afterwards, we illustrate how  these ideas extend to the mixed moments and cumulants of the GUE and then show how to  tweak our arguments to make them suitable for the GOE case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' At the end of this subsection,  we also discuss connections to topological and combinatorial (hyper)maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  154  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Ribbon graphs for the GUE spectral moments  Let H be drawn from the N × N Gaussian unitary ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' According to the convention  established in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, this means that the entries {Hij}N  i,j=1 of the complex Hermitian  matrix H are centred normal variables with covariances given by  �  HijHkl  � = 1  2χk=jχl=i,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  where we recall that the indicator function χA equals one when A is true and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Here and throughout the remainder of this subsection, we suppress the fact that our averages  are with respect to the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(G)(H) given in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our first order of business is  to use the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) to give a multi-sum expression for the GUE spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let k ∈ N, recall the definition of Pk from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, and set ik+1 := i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The spectral  moment m(GUE)  k  is zero if k is odd, while for even k,  m(GUE)  k  = 2−k/2 ∑  σ∈Pk  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ik=1  (χiσ(1)=iσ(2)+1χiσ(2)=iσ(1)+1) · · · (χiσ(k−1)=iσ(k)+1χiσ(k)=iσ(k−1)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) as our definition for m(GUE)  k  , it is well known that  m(GUE)  k  =  �  Tr Hk�  =  �  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ik=1  Hi1i2Hi2i3 · · · Hik−1ik Hiki1  �  =  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ik=1  �  Hi1i2Hi2i3 · · · Hik−1ik Hiki1  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  the second line can be checked via straightforward induction, while the third line follows  from linearity of the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The summand on the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) is  the expected value of a product of normally distributed random variables, so the Isserlis–Wick  theorem applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, it vanishes for odd k, and for even k, we have  �  Hi1i2Hi2i3 · · · Hik−1ik Hiki1  �  = ∑  σ∈Pk  �  Hiσ(1)iσ(1)+1Hiσ(2)iσ(2)+1  �  · ·  �  Hiσ(k−1)iσ(k−1)+1Hiσ(k)iσ(k)+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  Substituting this expression into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), interchanging the order of summation, and  using equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) then produces the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  155  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  The inner sum in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) reduces to ∑N  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',jl=1 1 = Nl for some 1 ⩽ l ⩽ k/2 + 1  since many of the indices i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ik must be identified to ensure that the product of indicator  functions is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To better understand this statement, it is convenient to recast it in  terms of ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This can be done in a natural way by recognising that each choice  of σ ∈ Pk determines a partitioning of the set {Hi1i2, Hi2i3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Hiki1} into k disjoint pairs: To  begin, we encode the fact that the first index of each element of {Hi1i2, Hi2i3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Hiki1} is equal  to the second index of one other element of the same set by drawing a k-gon whose vertices  are labelled i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ik in clockwise order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' each edge of the k-gon can be oriented in two  ways, with (il → il+1) representing the matrix entry Hilil+1 and (il+1 → il) representing Hilil+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Next, for each l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k/2, we represent the identification of the oriented polygon edges  (iσ(2l−1) → iσ(2l−1)+1) and (iσ(2l)+1 → iσ(2l)), implied by the factor χiσ(2l−1)=iσ(2l)+1χiσ(2l)=iσ(2l−1)+1  in the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), by connecting them with an untwisted ribbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4: These are the ribbon graph representations of the terms given in equations  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The purple (grey) ribbons represent the first (second) factor in the  summands of the corresponding equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us illustrate the above formalism in the k = 4 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The set P4 contains  three permutations: the identity, (2, 4, 3), and (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) tells us that  m(GUE)  4  = 1  4  �  Sid + S(2,4,3) + S(2,3)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  with  Sid := 4  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i4=1  ⟨Hi1i2Hi2i3⟩ ⟨Hi3i4Hi4i1⟩ =  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i4=1  (χi1=i3χi2=i2)(χi3=i1χi4=i4),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  S(2,4,3) := 4  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i4=1  ⟨Hi1i2Hi4i1⟩ ⟨Hi2i3Hi3i4⟩ =  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i4=1  (χi1=i1χi4=i2)(χi2=i4χi3=i3),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  S(2,3) := 4  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i4=1  ⟨Hi1i2Hi3i4⟩ ⟨Hi2i3Hi4i1⟩ =  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i4=1  (χi1=i4χi3=i2)(χi2=i1χi4=i3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  156  i2  11  i1  i2  1  i2  14  14  13  i4  i3  (2,3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  These sums are then represented by the ribbon graphs in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Seen as two-dimensional  surfaces, they have, respectively, three, three, and one boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This reflects the fact that  i3 ≡ i1 in the first ribbon graph, i4 ≡ i2 in the second ribbon graph, and i4 ≡ i3 ≡ i2 ≡ i1 in  the third ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence,  Sid =  N  ∑  i1,i2,i4=1  1 = N3,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  S(2,4,3) =  N  ∑  i1,i2,i3=1  1 = N3,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  S(2,3) =  N  ∑  i1=1  1 = N,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  so that by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), m(GUE)  4  = N3/2 + N/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The computation of the above example extends quite simply to the case of k being a  general positive even integer: One need only interpret Pk within Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 as the set of all  orientable (k/2)-ribbon graphs that can be built from a single k-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let k ∈ 2N and Pk be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, we have  m(GUE)  k  = 2−k/2 ∑  σ∈Pk  NV(σ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  where V(σ) is equal to the number of boundaries of the ribbon graph constructed from σ according to  the prescription above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Inspection of the ribbon graph corresponding to σ shows that each indicator function  in the summand of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) is represented by the side of a ribbon identifying two  vertices of the k-gon at the centre of the ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the number of unique summation  indices i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ik is equal to V(σ), the number of boundaries of said ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, we  see the reduction  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ik=1  (χiσ(1)=iσ(2)+1χiσ(2)=iσ(1)+1) · · · (χiσ(k−1)=iσ(k)+1χiσ(k)=iσ(k−1)+1) =  N  ∑  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',jV(σ)=1  1 = NV(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Substituting this into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) produces the sought result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling that ribbons represent edge identifications of the underlying k-gons,  V(σ) is also equal to the number of distinct vertices retained by a k-gon after identifying  edges according to the ribbon graph corresponding to σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  157  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5: These compact orientable surfaces (sphere, sphere, and torus) are obtained by  enlargening the white squares in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 while collapsing the ribbons therein to their  now-identified ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Three, three, and one distinct vertices survive the edge identification  procedure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It is evident from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) that m(GUE)  k  is a polynomial in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, according  to the statement (which we are yet to fully justify) following the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, it is a  polynomial of degree k/2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This means that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) can be rewritten as  m(GUE)  k  = 2−k/2  k/2+1  ∑  V=1  NV #{σ ∈ Pk | V(σ) = V},  k ∈ 2N,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  where #S denotes the number of elements in the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With our current definitions of the  function V(σ), computing the coefficient of NV on the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  requires us to generate all orientable (k/2)-ribbon graphs that can be built from a k-gon  and then count how many of them have exactly V boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we are able to  give a much neater interpretation of these polynomial coefficients, while also justifying our  assumption that the degree of m(GUE)  k  in N is k/2 + 1, by classifying our ribbon graphs with  respect to the genera of the corresponding compact surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us proceed by first observing that rather than collapsing the ribbons of a ribbon graph  (as described in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), another way of obtaining the compact surface represented by a  connected ribbon graph is to glue open disks along the boundaries of said ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  is seen to be true by noting that contracting the disks in the resulting polygonised surface to  points while also collapsing the ribbons to edges is equivalent to just collapsing the ribbons  in the original ribbon graph (see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, connected ribbon graphs can  be drawn on the compact surfaces that they represent and, moreover, these surfaces are of  minimal (Euler) genus such that this is possible without self-intersections (if one were able  to draw a ribbon graph on a surface of lower (Euler) genus without self-intersections, then  158  13  S(2,3)  Sid3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  shrinking the ribbons and disks bounded by the ribbons in the described way could not  produce the compact surface that the ribbon graph represents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As one would expect, this  means that the (Euler) genus of a ribbon graph (recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) is equal to that of the  compact surface it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6: Gluing a pink, green, and blue disk to the boundaries of the ribbon graph  illustrated on the left produces the polygonised sphere in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Shrinking the ribbons  to edges and (deformed) disks to vertices while enlargening the white square produces the  illustration on the right, which is exactly the first surface presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Recall that the Euler characteristic of a compact surface is given by the classical formula  χ = F − E + V,  where F, E, V are respectively the number of faces, edges, and vertices of a chosen poly-  gonisation of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In our case, the surfaces of interest are constructed by gluing  together pairs of edges of a k-gon, so an obvious choice of polygonisation for a given surface  is the edge-identified k-gon that it is equivalent to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we have F = 1 and E = k/2 since  our k-gons have one face and k edges which are identified pairwise into k/2 distinct edges  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Combining this with Euler’s formula and noting that the Euler characteristic  of a compact orientable surface is also given by χ = 2 − 2g, where g is the genus of the  surface, we have  V = 1 + k/2 − 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  Thus, we may reformulate equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) as a genus expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let k ∈ N and let Pk be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If k is odd, the spectral moment  m(GUE)  k  is zero, while if k is even, we have that  m(GUE)  k  = 2−k/2  k/2  ∑  l=0  N1+k/2−l #{σ ∈ Pk | g(σ) = l/2},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  where g(σ) is equal to the genus of the ribbon graph labelled by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  159  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 implies equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) with the possible requirement that the upper  terminal of the sum be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) and the discussion above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 tells  us that the coefficient of NV on the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) is equal to the number  of genus (1 + k/2 − V)/2, connected, orientable ribbon graphs that consist of a single k-gon  and k/2 ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, since g ⩾ 0, we see that V ⩽ k/2 + 1, which establishes that  the upper limit of the sum in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) is indeed k/2 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' it is a simple exercise to  check that the identity permutation σ = id ∈ Pk corresponds to a (genus zero) sphere, so  that V(id) = k/2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting l = 2g and changing the summation index in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  according to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) gives equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Replacing k by 2k in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) gives a proof of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='89) of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, in the GUE case and confirms our claim following Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that  2kM(GUE)  k,l  counts the number of orientable genus l/2 ribbon graphs that can be constructed  from a 2k-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, since g ∈ N and l = 2g, M(GUE)  k,l  = 0 for l odd and m(GUE)  2k  is consequently an odd polynomial in N, in keeping with the second point of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, let us point out that the interpretation of M(GUE)  k,l  in terms of ribbon graphs extends  to a representation of the generating functions W(GUE),l  1  (x) (recall the discussion at the  beginning of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) as genus l/2 compact orientable surfaces with one hole — if one can  replace the hole of such a representation of W(GUE),l  1  (x) by a 2k-gon for some positive integer  k and connect the edges of the 2k-gon with ribbons drawn on the surface without self-  intersections such that excising the resulting ribbon graph yields a disjoint union of sets  homeomorphic to open disks, then that ribbon graph contributes to the value of M(GUE)  k,l  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7: The sphere with a hole on the left represents W(GUE),0  1  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the right, we have  a collection of ribbon graphs drawn on this sphere with the hole replaced by a suitable  2k-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The illustrations on the right contribute to the value of M(GUE)  k,0  with k = 1, 2, 2, 3,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Therefore, they also contribute to the coefficients of the 1/x expansion of  W(GUE),0  1  (x) since the latter is essentially a generating function for the M(GUE)  k,0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  160  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) of ˜m(G)  2k , Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9 tells us that  ˜m(GUE)  2k  = 2−k  k  ∑  l=0  N−l #{σ ∈ P2k | g(σ) = l/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, from equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32),  ˜M(GUE)  2k,0  = limN→∞ ˜m(GUE)  2k  = M(GUE)  k,0  is equal to 2−k  multiplied by the number of genus zero (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', planar) ribbon graphs that can be built  from a 2k-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, from the discussion contained in the caption of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and  the paragraph preceding it, 2k ˜M(GUE)  2k,0  is equal to the kth Catalan number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Consequently,  W(GUE),0  1  (x), as described below equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19), is essentially a generating function  for the Catalan numbers, which is well known [290] to be given by  W(GUE),0  1  (x) = 1  2(x −  �  x2 − 2)  after appropriate scaling (see also differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Of course, applying  the Sokhotski–Plemelj inversion formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) yields the Wigner semi-circle law  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This proof of the Wigner semi-circle law is a rewording of Wigner’s original  proof [313], which used the so-called method of moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The above proof can be extended to show that the Wigner semi-circle law holds for the  Hermitian Wigner matrix ensemble, which is represented by the same matrices as the  GUE except that the moments ⟨Hp1+1  ij  Hp2+1  ji  ⟩ (p1, p2 ∈ N) no longer need to be zero  whenever p1, p2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Setting rigour aside (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [21, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2] for a detailed treatment),  the idea is that when H is a Hermitian Wigner matrix, the right-hand side of equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) needs to be replaced by a sum over all partitions of k, rather than just pair  partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As an example, the contribution of the terms ⟨Hi1i2Hi2i3⟩ ⟨Hi3i4Hi4i5Hi5i6Hi6i1⟩  and ⟨Hi1i2Hi2i3Hi3i4⟩ ⟨Hi4i5Hi5i6Hi6i1⟩ to the value of ⟨Hi1i2Hi2i3 · · · Hi5i6Hi6i1⟩ must now  be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, since these terms are interpreted as products of  indicator functions which constrain our summation indices i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i6, only pair partitions  contribute at leading order — ⟨Hi1i2Hi2i3Hi3i4Hi4i5⟩ is non-zero only if i3, i4, i5 depend  on i1, i2 while ⟨Hi1i2Hi2i3⟩ ⟨Hi3i4Hi4i5⟩ being non-zero merely constrains i3, i5 to depend  on i1, i2, i4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, at leading order in N, the spectral moments of the Hermitian Wigner  matrix H are given by the Catalan numbers up to some normalisation factor, and the  argument given above follows through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  161  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Ribbon graphs for the GUE mixed moments and cumulants  Letting H be again drawn from the N × N GUE, the mixed moments of the GUE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) are  given by  m(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  n  ∏  i=1  Tr Hki  �  ,  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The analogue of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) is then  m(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  N  ∑  i(1)  1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i(1)  k1 =1  · ·  N  ∑  i(n)  1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i(n)  kn =1  ��  Hi(1)  1 i(1)  2 Hi(1)  2 i(1)  3 · · · Hi(1)  k1 i(1)  1  �  · ·  · ·  �  Hi(n)  1 i(n)  2 Hi(n)  2 i(n)  3 · · · Hi(n)  kn i(n)  1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  The summand on the right-hand side of this equation is the expected value of a product  of centred normal variables, so it can be simplified using the Isserlis–Wick theorem, in a  similar manner to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, m(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanishes if k1 + · · · + kn is an odd integer,  whereas for k1 + · · · + kn an even integer, we can express m(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn in terms of ribbon graphs:  For each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n, the set {Hi(j)  1 i(j)  2 , Hi(j)  2 i(j)  3 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Hi(j)  kj i(j)  1 } is represented by a kj-gon with  vertices labelled i(j)  1 , i(j)  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(j)  kj in clockwise order, while the factors ⟨Hi(j)  p i(j)  p+1Hi(l)  q i(l)  q+1⟩ (setting  i(j)  kj+1 := i(j)  1 ) in the sum of products of covariances produced by the Isserlis–Wick theorem  are represented by untwisted ribbons connecting the appropriate polygon edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8: In computing m(GUE)  4,3,3  =  �  Tr(H4) Tr(H3) Tr(H3)  �  , one first draws a square and  two triangles with vertices labelled as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, one must list out all possible ways of  pairing the polygon edges using untwisted ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The illustrated planar ribbon graph  represents ⟨Hi(1)  1 i(1)  2  Hi(1)  2 i(1)  3 ⟩⟨Hi(1)  3 i(1)  4  Hi(1)  4 i(1)  1 ⟩⟨Hi(2)  2 i(2)  3  Hi(2)  3 i(2)  1 ⟩⟨Hi(2)  1 i(2)  2  Hi(3)  3 i(3)  1 ⟩⟨Hi(3)  1 i(3)  2  Hi(3)  2 i(3)  3 ⟩,  which, when summed over the indices, contributes a value of N6 to m(GUE)  4,3,3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As a slight abuse of notation, let us now define Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, for positive integers k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn  such that k1 + · · · + kn is even, to be the set of all 1  2(k1 + · · · + kn)-ribbon graphs that can be  162  z(1  2  1  1  (3)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  constructed by drawing n polygons with respectively k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn edges and then connecting  said edges with untwisted ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) extends to the form  m(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2− 1  2 (k1+···+kn)  1  2 (k1+···+kn)+n  ∑  V=1  NV #{Γ ∈ Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | V(Γ) = V},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  where V(Γ) is equal to the number of boundaries of Γ, which is equivalent to the number  of distinct polygon vertices retained by Γ upon collapsing its ribbons in order to identify  their ends (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 above, collapsing the ribbons `a la Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 results in there  being six distinct vertices labelled by i(1)  1  ≡ i(1)  3 , i(1)  2 , i(1)  4 , i(2)  1  ≡ i(2)  2  ≡ i(3)  1  ≡ i(3)  3 , i(2)  3  and i(3)  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  When we discussed the n = 1 case earlier, we saw that the equivalent function V(σ) could  be expressed in terms of topological invariants of the ribbon graph corresponding to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We  would now like to see if such relations can be replicated in the general-n case and moreover  justify the upper limit of the sum in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let Γ ∈ Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and let Σ denote the surface obtained by collapsing the ribbons of Γ such  that the ribbon-ends are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As in the n = 1 case, the Euler characteristic of Σ is given  by the classical formula χ = F − E + V, where F, E, V are again the number of faces, edges,  and vertices of any polygonisation of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Polygonising Σ by the polygons of Γ with their  edges identified according to the ribbon data of Γ, we have F = n, the number of polygons  in Γ, and E = (k1 + · · · + kn)/2, half the total number of polygon edges in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, the  number of vertices, V = V(Γ), is given by  V(Γ) = χ + 1  2(k1 + · · · + kn) − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  Writing C(Σ) for the number of connected components of Σ and gi ⩾ 0 for the genus of  the ith such component, we have by the additive property of the Euler characteristic that  χ = ∑  C(Σ)  i=1 (2 − 2gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It follows that χ ⩽ 2n since 1 ⩽ C(Σ) ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Combined with the above  equation, this means that V(Γ) ⩽ 1  2(k1 + · · · + kn) + n, which is precisely the upper limit  of the sum in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, unlike in the n = 1 case, this upper limit cannot  always be attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, if any of the ki are odd, the corresponding polygon must be  ribbon-connected to at least one other polygon, so Γ, equivalently Σ, cannot have n connected  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This slight complication in determining the degree of m(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn as a polynomial  in N, in addition to the reasons listed below, suggests that we should somehow focus on  studying connected ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  163  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If we define  Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn := {Γ ∈ Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | Γ is connected},  we have by the arguments above that for all Γ ∈ Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  V(Γ) ⩽ 1  2(k1 + · · · + kn) + 2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Furthermore, this relation is a strict equality if Γ is planar (genus zero) — it is easy to  convince oneself that there exists at least one such Γ in each Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Connected surfaces have well-defined genera, so we can rewrite the analogue of  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) with Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn replaced by Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn as a genus expansion, in a similar  fashion to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' No generality is lost by studying connected ribbon graphs as every ribbon graph is a  countable disjoint union of its connected components, which is easy to construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To expand on the third point given above, it is well known to combinatorialists [290, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5]  that if the mixed moment mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn on the left-hand side of the moment-cumulants relation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) is the weighted count of some structures of a given type (graphs, marked surfaces,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') built from n objects, then the corresponding cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn are the analogous counts  of the subsets of such structures that are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This can be understood in the context  of ribbon graphs from the following inductive argument: If we assume for all p < n and  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kp ∈ N such that k1 + · · · + kp is an even integer that the mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kp  enumerate (with suitable weights) the number of connected ribbon graphs that can be built  from p polygons with respectively k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kp sides, then the product ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kpcl1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',lq (q < n)  counts the number of ribbon graphs consisting of two connected components with the first  being a ribbon graph built from p polygons with k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kp sides and the second component  being a ribbon graph built from q polygons with l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , lq sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Extending this reasoning to  general products, we see that the sum  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn − ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  ∑  K⊢{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn}  K̸={{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn}}  ∏  κi∈K  cκi  counts the number of disconnected ribbon graphs that can be constructed from n polygons  with k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn enumerates both the disconnected and connected ribbon  164  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  graphs that can be built from n polygons with k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn sides, it follows that ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn must  enumerate all of the connected ribbon graphs that can be constructed in said manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanishes whenever k1 + · · · + kn is odd, so too does ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This can  be seen from the inverse of the moment-cumulants relation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)),  ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  ∑  K⊢{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn}  (−1)#K−1(#K − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ∏  κi∈K  mκi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  Having established that we are interested in enumerating connected, orientable ribbon  graphs and that these enumerations are given by the mixed cumulants of the GUE, let us  now give the general-n analogue of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N be such that k1 + · · · + kn is even and let Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the  set of connected, orientable 1  2(k1 + · · · + kn)-ribbon graphs built from n polygons with respectively  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The mixed cumulant c(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is given by  c(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  � N  2  � 1  2 (k1+···+kn) 1  2 (k1+···+kn)+1−n  ∑  l=0  N2−n−l #{Γ ∈ Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | g(Γ) = l/2},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  where g(Γ) is the genus of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First, note that if Γ ∈ Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, it is connected with Euler characteristic χ = 2 − 2g, so  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) reads  V(Γ) = 1  2(k1 + · · · + kn) + 2 − n − 2g ⩽ 1  2(k1 + · · · + kn) + 2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  Thus, the analogue of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) corresponding to connected ribbon graphs is  c(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2− 1  2 (k1+···+kn)  1  2 (k1+···+kn)+2−n  ∑  V=1  NV #{Γ ∈ Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | V(Γ) = V}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  Setting l = 2g and changing summation index according to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) in equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In comparing equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), we see that a benefit of studying the mixed  cumulants is that upon scaling them by a factor of N−(k1+···+kn)/2 (equivalent to considering  the cumulants of the GUE scaled by replacing the matrix H in P(G)(H) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) by  √  NH as is  consistent with Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), they are O(N2−n), while the mixed moments are O(Nn) when  165  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  scaled in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the connected n-point correlators ˜W(GUE)  n  of the global scaled  GUE, which are generating functions for the scaled cumulants, have large N expansions of  the form given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, whereas the corresponding unconnected n-point correlators  ˜U(GUE)  n  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In keeping with the discussion surrounding Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7, let us also  mention that it is convenient to represent the correlator expansion coefficients W(GUE),l  n  defined in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 as genus l/2 compact, connected, orientable surfaces with n holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Ribbon graphs in the GOE case  The calculations pertaining to the GUE that have been outlined thus far in this subsection  follow through in much the same way when H is drawn from the N × N Gaussian orthogonal  ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, when H is an N × N GOE matrix, the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  now equals m(GOE)  k  with the summand still simplifying to the form given in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  when k is a positive even integer (and vanishing if k is odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The key difference in studying  the GOE is that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) must be replaced with  �  HijHkl  � = 1  4  �  χk=jχl=i + χk=iχl=j  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  which reflects the fact that H is now a real symmetric matrix with off-diagonal entries having  variance 1/4 (when i ̸= j, ⟨H2  ij⟩ =  �  HijHji  � = 1/4) and diagonal entries having variance 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To ease notation, let us define  ξkl  ij (t) := (1 − t)χk=jχl=i + tχk=iχl=j  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  so that the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) is equal to 1  4[ξkl  ij (0) + ξkl  ij (1)] (the parameter  t can be thought of as keeping track of orientability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the role of γ in [166]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  substituting equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) shows that for k a positive even integer,  �  Hi1i2Hi2i3 · · · Hik−1ik Hiki1  � = 2−k ∑  σ∈Pk  1  ∑  t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',tk/2=0  ξ  iσ(2)iσ(2)+1  iσ(1)iσ(1)+1(t1) · · · ξ  iσ(k)iσ(k)+1  iσ(k−1)iσ(k−1)+1(tk/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  As in the GUE case, we now encode the index-matching of the set {Hi1i2, Hi2i3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Hiki1} by  drawing a k-gon with vertices labelled i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ik in clockwise order, and then represent  the edge-matching implied by the summand of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) by drawing k/2 ribbons  connecting the edges of said k-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The factors ξ  iσ(2l)iσ(2l)+1  iσ(2l−1)iσ(2l−1)+1(tl) (1 ⩽ l ⩽ k/2) in the  166  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) are equivalent to the products χiσ(2l−1)=iσ(2l)+1χiσ(2l)=iσ(2l−1)+1  seen in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) when we set tl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the term ξ  iσ(2l)iσ(2l)+1  iσ(2l−1)iσ(2l−1)+1(0) is represented  by an untwisted ribbon connecting the oriented polygon edges (iσ(2l−1) → iσ(2l−1)+1) and  (iσ(2l)+1 → iσ(2l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, the term ξ  iσ(2l)iσ(2l)+1  iσ(2l−1)iσ(2l−1)+1(1) is non-zero if the edge  (iσ(2l−1) → iσ(2l−1)+1) is identified to (iσ(2l) → iσ(2l)+1), so we represent it by a M¨obius  half-twisted ribbon connecting these two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9: These ribbon graphs represent the term ξi2i3  i1i2(t1)ξi4i1  i3i4(t2) for various choices of  t1, t2 ∈ {0, 1}, with the purple (grey) ribbons representing the first (second) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In order,  the graphs correspond to setting (t1, t2) = (0, 1), (1, 0), and (1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' they can be obtained by  twisting either or both of the ribbons of the left image of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, which incidentally  corresponds to setting (t1, t2) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Extending the above reasoning to products of the form presented in the summand of  the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) shows that computing the mixed moments and  cumulants of the GOE amounts to enumerating locally orientable ribbon graphs of specific  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The necessary arguments for obtaining explicit expressions for these moments and  cumulants are the same as in the GUE case, with the key numeric still being the number of  boundaries of each ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Without reiterating any further details, let us simply state  the GOE analogue of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N be such that k1 + · · · + kn is even and let ˜Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the set  of all connected, locally orientable 1  2(k1 + · · · + kn)-ribbon graphs that can be built from n polygons  with respectively k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The mixed cumulant c(GOE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is a degree 1  2(k1 + · · · + kn) + 2 − n  polynomial in N given by the formula  c(GOE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  � N  4  � 1  2 (k1+···+kn) 1  2 (k1+···+kn)+1−n  ∑  l=0  N2−n−l #{Γ ∈ ˜Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | ˜g(Γ) = l},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  where ˜g(Γ) is the Euler genus of Γ (recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  167  i1  i1  i2  i4 3  i3  i4  i3  i4CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The set ˜Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn defined above is in bijection with Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn × {0, 1}(k1+···+kn)/2  since Γ ∈ ˜Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn if and only if it is the result of twisting the ribbons of some ribbon graph  drawn from Pc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the tuple (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , t(k1+···+kn)/2) ∈ {0, 1}(k1+···+kn)/2 determines which  ribbons are to be twisted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Upon setting n = 1, comparing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) confirms that M(GOE)  k,l  has the combinatorial interpretation described in the paragraph below Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 (as an  aside, observe that the ribbon graph in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 is genus zero and thus contributes a value  of N2 to the computation of c(GOE)  1,2,3  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, the equality  {Γ ∈ ˜Pc  k | ˜g(Γ) = 0} = {Γ ∈ Pc  k | g(Γ) = 0}  implies that the moment expansion coefficients M(GUE)  k,0  and M(GOE)  k,0  differ only by a factor  of 2k/2, so the proof of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 can be used to show that the global scaled eigenvalue  density of the GOE is given by the Wigner semi-circle law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A key difference between c(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and c(GOE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is that, since the (Euler) genus is a non-  negative integer, the summand of the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) for c(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is identically zero for  odd values of l, but this is not true for the corresponding expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) for c(GOE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Hence, unlike in the GUE case, the correlator expansion coefficients W(GOE),l  n  do not vanish  for odd values of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another difference between W(GUE),l  n  and W(GOE),l  n  is that it does not  make sense to represent the latter as a compact surface on which to draw ribbon graphs (like  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) since it is a generating function for cumulants that count both orientable and  non-orientable ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Relations to topological and combinatorial maps and hypermaps  It is oftentimes beneficial, either for visualisation or computational purposes, to consider  alternative representations of ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We now outline the connection between ribbon  graphs and two such representations: topological and combinatorial maps [303], [220], [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A topological map is a graph embedded in a compact surface such that the  edges of the graph do not intersect and excising the graph from the surface results in a  disjoint union of open sets that are homeomorphic to open disks — these open sets are  referred to as the faces of the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  168  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Note 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The second condition in the above definition requires that for a given embedding to  be a valid topological map, each connected component of the graph must live on a separate  connected component of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Most authors require topological maps, hence the  involved graphs, to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In such settings, it suffices to define a topological map as  the embedding of a graph in a connected compact surface of minimal (Euler) genus such  that the edges of the graph do not intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  There are two natural ways of transforming a ribbon graph into a topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In  the first formalism, one glues open disks to the boundaries of a given ribbon graph and  then contracts the polygons of the ribbon graph to points which become the vertices of a  topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Shrinking the polygons to vertices also has the consequence of thinning the  ribbons to edges connecting said vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The disks that were initially glued to the ribbon  graph become the faces of the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10: As in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6, we glue three open disks (pink, green, and blue) to the  boundaries of the ribbon graph on the left in order to embed it into a sphere (middle image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Shrinking the white square to a vertex and the ribbons to edges incident to said vertex  produces the topological map on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our usual convention, which we break here,  will be to colour the vertices black and the faces white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the second formalism, one also glues disks to the boundaries of the given ribbon graph,  but then contracts these disks to vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The ribbons then collapse to their ends, which  become the edges of the topological map, and the polygons of the ribbon graph become  the faces of the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This construction was already exemplified in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The two formalisms just described lead to topological maps that are dual to each other in  the sense that canonically follows from the duality principle well known in standard graph  theory: one can be obtained from the other by placing a vertex at the centre of every face,  drawing an edge between each pair of new vertices whenever the corresponding faces of  the original topological map share an edge, and then deleting the edges and vertices of the  169  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  original topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, the topological maps of Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 are dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For the remainder of this subsection, when we refer to the topological map corresponding to  a ribbon graph, we will mean the one constructed via the first formalism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', that illustrated  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It is important to observe at this point that neither of the above constructions are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For example, the first two ribbon graphs displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, which can be distinguished  due to the labelling of the polygon vertices (that we often leave implicit), produce the same  topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case of orientable ribbon graphs, we are able to turn the first  construction into a bijection by additionally marking a half-edge of each of the n vertices in  a systematic manner, thereby forming a so-called n-rooted topological map [163].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One possible  convention is to mark the half-edges corresponding to the ribbon-ends (i(l)  1  → i(l)  2 ) and label  them by l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see [219], [277] for more comprehensive reviews of (1-)rooted topological maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11: These rooted topological maps are constructed from the ribbon graphs of  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 using the first prescribed formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The half-edges that are root-marked red  correspond to the ribbon ends that are glued to the polygon edges (i1 → i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the case of locally orientable ribbon graphs, one also needs to keep track of local  orientation at the vertices and which edges of a given topological map correspond to twisted  ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This can be done by assigning orientation to the half-edge root-markings and  labelling each edge by, say, ±1, with +1 indicating that the edge represents an untwisted  ribbon and −1 instead referring to a M¨obius half-twisted ribbon (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [252]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Seeing as how  introducing markings and labellings to give a combinatorial flavour to our topological  maps makes the theory more tractable, one might wonder if much more can be gained  by introducing even more labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that it is possible to transition to a fully  combinatorial theory, removing the need for explicitly constructing graphs or surfaces  altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we are led to combinatorial maps, for which there are multiple definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  170  1  03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 (Tutte ’84).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following [303], [161, §17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10], a combinatorial map is a quadruple  (EQ, τ0, τ1, τ2) where  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' EQ is a finite set with the number of elements being divisible by four,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' τ0, τ1, τ2 are fixed-point free involutions on EQ,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' τ0τ1 = τ1τ0 and τ0τ1 is also fixed-point free,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the group ⟨τ0, τ1, τ2⟩ generated by τ0, τ1, τ2 is transitive, meaning that it has exactly one  orbit, that being all of EQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The elements of EQ are called quarter-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The orbits of τ0, ⟨τ0, τ1⟩, ⟨τ0, τ2⟩, and ⟨τ1, τ2⟩ are  respectively referred to as the half-edges, edges, vertices, and faces of the combinatorial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Combinatorial maps are in one-to-one correspondence with connected, n-rooted topologi-  cal maps that have been edge-labelled ±1 in the aforementioned manner (the connectedness  being enforced by the fourth condition in the above definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This correspondence follows  from the suggestive names given to the elements of EQ and the various orbits highlighted  in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3: Given a suitable topological map, one must first divide each edge both  width- and length-wise to form quarter-edges and then assign these quarter-edges distinct  labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the corresponding combinatorial map has for EQ the set of said quarter-edges,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  while the involutions τ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' τ2 on EQ are such that τ0 interchanges two quarter-edges if and  only if they belong to the same half-edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' τ1 interchanges two quarter-edges if and only if  they belong to the same length-wise half-edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and τ2 interchanges two quarter-edges if and  only if they are incident to the same vertex without belonging to the same half-edge and are  consecutive to each other in the cyclic ordering around that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The two possible choices  of (local) orientation at the vertices of the topological map are encoded in the cycles of τ0τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The combinatorial map corresponding to Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12 below is (EQ, τ0, τ1, τ2)  with  EQ = {1, 1′, 2, 2′, 3, 3′, 4, 4′},  τ0 = (1, 1′)(2, 2′)(3, 3′)(4, 4′),  τ1 = (1, 2′)(1′, 2)(3, 4′)(3′, 4),  τ2 = (1, 4′)(1′, 2)(2′, 3)(3′, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Observe that the first cycle of τ0τ2 = (1, 4, 3, 2)(1′, 2′, 3′, 4′) describes an anticlockwise order-  ing of the half-edges incident to the single vertex of the topological map, while the second  171  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  cycle of τ0τ2 describes the opposite ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These cyclic orderings each prescribe a choice  of orientation for the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12: We divide the ribbons of a ribbon graph (left) and edges of the corresponding  topological map (right) into quarters that are labelled 1, 1′, 2, 2′, 3, 3′, 4, 4′ in clockwise order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The half-edge marked red and oriented clockwise corresponds to the ribbon-end (i1 → i2),  while the blue plus signs signify that these edges represent untwisted ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  A combinatorial map is (globally) orientable if and only if ⟨τ0τ1, τ0τ2⟩ has two distinct  orbits [303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If this is the case, one may remove the redundancy of τ0 by identifying quarter-  edges whenever they belong to the same half-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the context of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12, this amounts  to setting j′ ≡ j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Identifying quarter-edges according to τ0 induces a choice of  orientation and moreover leads to a simpler combinatorial theory on the set of half-edges  EH ≃ EQ/τ0, which has half as many elements as that considered in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  simplification is to be expected since every compact surface has an orientable double cover  [176], so a theory allowing for non-orientable objects should intuitively require twice as  many labels as an equivalent theory focusing solely on orientable structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 (Edmonds ’60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following [101], [220], an oriented combinatorial map is a triple  (EH, τe, τv) where  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' EH is a finite set with an even number of elements,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' τv is a permutation of EH and τe is a fixed-point free involution on EH,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the group ⟨τe, τv⟩ is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The elements of EH are called half-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The cycles of τe, τv, and τ−1  v τ−1  e  are respectively  called the edges, vertices, and faces of the oriented combinatorial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content="  172  :1  i2  2  4'  4  2'  2  i4  4  i3  3'  33." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  This definition is preferred to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 in the orientable case because the correspon-  dence between oriented combinatorial maps and orientable, connected, n-rooted topological  maps is somehow more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, if one labels the half-edges of such a topological  map and settles on a choice of orientation, then the corresponding oriented combinatorial  map has EH being the set of labelled half-edges, τe being the involution that interchanges  half-edges that belong to the same edge, and τv being a product of disjoint cycles where each  cycle describes the cyclic ordering of the half-edges incident to a vertex that is induced by  the orientation of the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Each cycle of the face permutation τ−1  v τ−1  e  describes  a face of the topological map by listing out the first half of each edge that borders the face  when traversing in the direction determined by the choice of orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13: Illustrated is an oriented planar graph with labelled half-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It should be  thought of as being embedded in the sphere and thus equivalent to an oriented topological  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The corresponding oriented combinatorial map is (EH, τe, τv) with EH = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 8},  τe = (1, 2)(3, 5)(4, 6)(7, 8), and τv = (1, 8)(2, 3, 4)(5, 7, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that the cycles of τv comply  with the orientation of the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, the first, second, and third cycle  of the face permutation τ−1  v τ−1  e  = (1, 4, 7)(2, 8, 5)(3, 6) respectively list out, in appropriate  cyclic order, the blue, black, and yellow halves of the edges bordering the triangular face,  the ‘external face’, and the bigonal face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us now briefly review the connection between the structures discussed above and  their hypermap analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A topological hypermap is simply a connected topological map  whose vertices are coloured black and white (later, we use red) in such a way that each  edge connects a black vertex to a white one [220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In that vein, a combinatorial hypermap is  the triple of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 except that τe is now free to be any permutation of EH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' EH is  then the set of edges of the hypermap, while the cycles of τv (τe) are interpreted as black  vertices (white vertices or so-called hyperedges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, an oriented combinatorial map is a  173  3  2  4  1  8  6  7  5CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  combinatorial hypermap with τe constrained to be a fixed-point free involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similarly, a  topological hypermap with all white vertices constrained to be bivalent is equivalent to a  connected topological map — the edges of the hypermap are interpreted as half-edges of the  topological map so that each white vertex sits at the middle of a map-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the literature surrounding Grothendieck’s dessins d’enfants [284], a topological  hypermap is referred to as a dessin d’enfant, while a topological map — seen as a topological  hypermap with bivalent white vertices — is referred to as a clean dessin d’enfant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Moments of the Laguerre unitary and orthogonal ensembles  Since the Laguerre unitary and orthogonal ensembles are represented by matrices that  can be expressed as products of Ginibre matrices, whose entries are normally distributed,  their (mixed) moments and cumulants can also be studied using the Isserlis–Wick theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Assuming mastery over the Gaussian case, as outlined in the previous subsection, we first  show how the spectral moments m(LUE)  k  can be written as sums over certain bicoloured  ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, as in the Gaussian case, we discuss how the presented ideas extend to  the mixed cumulants of the LUE and also the LOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Ribbon graphs for the LUE spectral moments  Let G be drawn from the M × N complex Ginibre ensemble and throughout this subsection,  let ⟨ · ⟩ denote averages with respect to the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(Gin)(G) given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' According  to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, the N × N Wishart–Laguerre matrix W = G†G then represents the (M, N)  Laguerre unitary ensemble and the spectral moments of this ensemble are given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  m(LUE)  k  =  �  Tr Wk�  ,  k ∈ N  =  �  Tr (G†G)k�  =  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ik=1  M  ∑  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',jk=1  �  (G†)i1j1Gj1i2(G†)i2j2Gj2i3 · · · (G†)ikjkGjki1  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  in analogy with equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since the real components of the entries of G are indepen-  dent, centred, normal variables such that  �  (G†)ijGkl  �  = χi=lχj=k,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  174  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  the Isserlis–Wick theorem then shows that  �  (G†)i1j1Gj1i2 · · · (G†)ikjkGjki1  �  = ∑  σ∈Sk  k  ∏  l=1  �  (G†)iljlGjσ(l)iσ(l)+1  �  = ∑  σ∈Sk  k  ∏  l=1  (χil=iσ(l)+1χjl=jσ(l)),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  where Sk is the set of all permutations of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k} and we have defined ik+1 := i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence,  combining equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) gives the analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 for the LUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Fix k ∈ N, let Sk be the set of permutations of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k} and set ik+1 := i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have  that the spectral moments of the LUE are given by  m(LUE)  k  = ∑  σ∈Sk  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ik=1  M  ∑  j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',jk=1  k  ∏  l=1  (χil=iσ(l)+1χjl=jσ(l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  As one might expect, the summand of the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) can be  represented by ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To do so, we first draw a 2k-gon and label its vertices  i1, j1, i2, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ik, jk in clockwise order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Next, to distinguish the two index sets, we colour  the vertices labelled by the i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ik black and those labelled by the j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , jk red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for  1 ⩽ l ⩽ k, edges of the form (il → jl) represent (G†)iljl, while those of the form (jl → il+1)  represent Gjlil+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The factors in the summand of the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) are  thus represented by untwisted ribbons connecting these two types of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The condition  that the ribbons cannot join edges of the same type is equivalent to that of only allowing  ribbon graphs that respect the colouring of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14: The illustrated ribbon graphs represent the terms given in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Having alternately coloured the vertices of the squares black and red  according to the formalism described above, we see that the vertex colourings induce  colours on the boundaries of the ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The number of black (red) boundaries is  equal to the exponent of N (M) in the evaluations of S′  id and S′  (1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  175  11  J1  12  i2  12  i2CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us now give the LUE analogue of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, wherein we saw that m(GUE)  4  is equal to (N3 + N3 + N)/4, with each of the terms N3, N corresponding to a ribbon graph  of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the LUE case, we compute the second spectral moment m(LUE)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' By equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31), it is given by  m(LUE)  2  = S′  id + S′  (1,2)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  with  S′  id :=  N  ∑  i1,i2=1  M  ∑  j1,j2=1  (χi1=i2χj1=j1)(χi2=i1χj2=j2) = M2N,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  S′  (1,2) :=  N  ∑  i1,i2=1  M  ∑  j1,j2=1  (χi1=i1χj1=j2)(χi2=i2χj2=j1) = MN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  The evaluation S′  id = M2N follows from the fact that the product of indicator functions  in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) enforces the equivalence i1 ≡ i2, but otherwise does not constrain the  indices j1, j2 — similarly for S′  (1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These equivalences of summation indices can be read off  from the ribbon graphs of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In general, the kth spectral moment for the LUE is given by  m(LUE)  k  = ∑  σ∈Sk  MVr(σ)NVb(σ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  where Vr(σ) and Vb(σ) are equal to the number of red, respectively black, boundaries of the  ribbon graph corresponding to σ — the proof of this fact is the same as for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As  with the GUE case, we are able to reformulate equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) as a genus expansion, so  long as the red and black vertices of the involved ribbon graphs are placed on equal footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let k ∈ N and Sk be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 Setting the Laguerre parameter a = ˆaN  with ˆa = O(1) so that M = (ˆa + 1)N, as per the convention set out in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we have  m(LUE)  k  =  k  ∑  l=0  N1+k−l  k  ∑  p=1  (ˆa + 1)p #{σ ∈ Sk | g(σ) = l/2 and Vr(σ) = p},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  where g(σ) is again the genus of the ribbon graph corresponding to σ — equivalently, g(σ) is  the minimal genus of the surface on which this ribbon graph can be embedded into without self-  intersections and it is also the genus of the surface obtained by pairwise identifying the edges of a  2k-gon according to the factors (χil=iσ(l)+1χjl=jσ(l)) in the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  176  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Observe that the construction outlined above Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 describes a bijection between  Sk and the set of orientable k-ribbon graphs built from 2k-gons with well-defined bicoloured  boundaries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', each ribbon has both a black and red side, and black (red) vertices lie on  black (red) boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, interpreting Sk as this latter set, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) can be  rewritten as  m(LUE)  k  =  k  ∑  Vr,Vb=1  MVr NVb #{σ ∈ Sk | Vr(σ) = Vr and Vb(σ) = Vb},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  where #S has the same meaning as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting in M = (ˆa + 1)N and  using equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) to relate V(σ) = Vr(σ) + Vb(σ), the total number of boundaries of  the ribbon graph labelled by σ, to g(σ) yields equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Comparing equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) shows that M(LUE)  k,l  has precisely the inter-  pretation described below Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Each ribbon graph that contributes to the computation  of m(LUE)  k  also contributes to the value of m(GUE)  2k  upon ‘forgetting’ the bicolouring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' compare  Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, not every LUE ribbon graph arises in this way, since some  orientable ribbon graphs have no valid bicolouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, colouring any vertex of  the third ribbon graph displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 red forces the single boundary and hence all  vertices to also be coloured red, which violates the bicolouring requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless,  every planar ribbon graph has a valid bicolouring, so that the genus zero ribbon graphs  pertaining to the calculation of m(GUE)  2k  are in one-to-one correspondence with the genus zero  bicoloured ribbon graphs contributing to the value of m(LUE)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In keeping with the discussion  following Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5, this implies the very simple relationship M(LUE)  k,0  = 2kM(GUE)  k,0  when  ˆa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In combination with the proof of the Wigner semi-circle law given in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we  thus have a combinatorial proof of the Marˇcenko–Pastur law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) in the regime a = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Ribbon graphs for the LUE and LOE mixed cumulants  The mixed moments m(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and cumulants c(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn of the LUE are respectively defined  through the formulae (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  m(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  n  ∏  i=1  Tr Wki  �  ,  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  =  ∑  K⊢{k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn} ∏  κi∈K  c(LUE)  κi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  177  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  The proofs of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) can be extended to give their LUE analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N and define Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to be the set of (k1 + · · · + kn)-ribbon  graphs that consist of n polygons with respectively 2k1, 2k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 2kn edges and vertices alternately  coloured black and red, whose edges are connected by untwisted ribbons that respect the bicolouring of  the polygon vertices — let Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the subset of connected ribbon graphs in Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, taking  a = ˆaN with ˆa = O(1), the mixed moments and cumulants of the LUE are given by  m(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  k1+···+kn+n  ∑  V=1  NV  ×  k1+···+kn  ∑  p=1  (ˆa + 1)p #{Γ ∈ Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | Vr(Γ) + Vb(Γ) = V and Vr(Γ) = p},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  c(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  k1+···+kn+1−n  ∑  l=0  Nk1+···+kn+2−n−l  ×  k1+···+kn+1−n  ∑  p=1  (ˆa + 1)p #{Γ ∈ Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | g(Γ) = l/2 and Vr(Γ) = p},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  where Vr(Γ) and Vb(Γ) are respectively the number of red and black boundaries of the ribbon graph Γ  and g(Γ) is the genus of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In contrast to what is seen in the GUE case, m(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is always guaranteed to be a  polynomial of degree k1 + · · · + kn + n in N since there is at least one Γ ∈ Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn such that  Vr(Γ) + Vb(Γ) equals this degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This Γ, which has n connected components that are each  of genus zero, exists because the polygons present in the ribbon graphs of Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn each have  an even number of sides, which allows for polygon edges to be paired without need for any  ribbons connecting two separate polygons — the GUE analogue of Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is the set Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  which contains no ribbon graphs with n connected components if any of the ki are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The (M, N) Laguerre orthogonal ensemble is represented by the N × N Wishart–Laguerre  matrix W = GTG, where G is an element of the M × N real Ginibre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Its (mixed)  moments and cumulants (which are specified by equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) with our new  definition of W) can be computed in the same way as in the LUE case, except that the involved  ribbon graphs are now allowed to have M¨obius half-twisted ribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is because G being  real means that (GT)ij = Gji, so equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) should be rewritten as  �  (GT)ijGkl  �  =  �  (GT)ij(GT)lk  �  =  �  GjiGkl  � = 1  2χi=lχj=k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  178  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  recalling that our ribbon graphs are built from polygons whose edges represent entries of  either GT or G, the first average in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) is represented by an untwisted ribbon  connecting these two types of edges, while the second (third) average therein is represented  by a M¨obius half-twisted ribbon connecting together two edges of the GT (G) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N and define ˜Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to be the set of locally orientable,  connected (k1 + · · · + kn)-ribbon graphs built from n polygons with respectively 2k1, 2k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 2kn  edges and vertices alternately coloured black and red, whose boundaries have well-defined colours  induced by the vertex colouring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, in the setting a = ˆaN with ˆa = O(1), we have  c(LOE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  k1+···+kn+1−n  ∑  l=0  � N  2  �k1+···+kn  N2−n−l  ×  k1+···+kn+1−n  ∑  p=1  (ˆa + 1)p #{Γ ∈ ˜Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | ˜g(Γ) = l and Vr(Γ) = p},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43)  where Vr(Γ) is the number of red boudnaries of Γ and ˜g(Γ) is the Euler genus of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note that the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) is very similar to that for c(LUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn given in equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41), with a key difference being that the summand of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) may be non-zero  for odd values of l, in contrast to what is seen in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, the mixed  moments of the LOE are given by a perturbation of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) involving locally  orientable ribbon graphs, though we do not display it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15: The illustrated bicoloured, locally orientable ribbon graph represents the  term ⟨(GT)i(1)  1 j(1)  1 (GT)i(2)  1 j(2)  1 ⟩⟨Gj(1)  1 i(1)  2  Gj(2)  2 i(2)  1 ⟩⟨(GT)i(1)  2 j(1)  2  Gj(1)  2 i(1)  1 ⟩⟨Gj(2)  1 i(2)  2 (GT)i(2)  2 j(2)  2 ⟩, where  we have adopted the vertex labelling convention of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Interpreting the ribbons  as identifying the polygon edges, we have two faces, four edges, and four vertices (equiv-  alently, boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, by the classical formula χ = F − E + V = 2 − ˜g, this ribbon  graph is an element of ˜Sc  2,2 with two red boundaries and Euler genus zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, by  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43), it contributes a value of (ˆa + 1)2N4 to c(LOE)  2,2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  179  R  i(1)  ji  i(2)  i2  i2  i记  12CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Like in the Gaussian case, the generating functions ˜W(LUE)  n  , ˜W(LOE)  n  of the corresponding  mixed cumulants scaled according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 are O(N2−n), so they have large N  expansions of the form given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another similarity between the GUE and LUE  is that the correlator expansion coefficients W(LUE),l  n  (recall their definition in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  and cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the discussion following equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)) can be visualised as genus l/2 compact,  connected, orientable surfaces with n holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This represents the fact that for given (n, l),  W(LUE),l  n  is a generating function for all of the ribbon graphs that can be drawn on such a  surface by replacing the holes with even-sided polygons and connecting the edges of these  polygons with ribbons such that the resulting (embedded) ribbon graph satisfies the criteria  given above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 (the ribbon graph does not self-intersect and excising it from the  surface results in a disjoint union of sets homeomorphic to open disks), along with the  condition that the disks bounded by the ribbons can be coherently bicoloured black and red  such that each ribbon borders both a black and red face — in the LOE case, one needs to be  careful and categorise the relevant ribbon graphs by their orientability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Relations to topological and combinatorial maps and hypermaps  As was discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, alternative representations of ribbon graphs are interesting for  various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The bicoloured ribbon graphs constructed above can be represented by  the topological maps introduced in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 in a similar fashion to that discussed  below said definition, with a few differences: Recall that a true bijection between sets of  ribbon graphs and their topological map counterparts requires that the latter be rooted (see  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) and that, in the locally orientable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', LOE) case, each vertex be assigned a local  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, in contrast to the GOE case, we no longer need to keep track of which  map edges correspond to twisted ribbons since this data can be recovered by checking if  the ends of a map edge correspond to polygon edges of the same or different type (recall  that the polygon edges of an LOE ribbon graph represent entries of either GT or G) — this is  assuming that the relevant topological maps are constructed using the formalism illustrated  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (As an aside, observe that there is a bijection between ˜Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and the set  Pc  2k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',2kn of connected, orientable GUE ribbon graphs: Alternately labelling the edges of  each polygon of Γ ∈ Pc  2k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',2kn by GT and G, with the ‘first’ or rooted edge of each polygon  180  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  labelled GT, and then twisting any ribbons connecting polygon edges of the same type results  in a ribbon graph that can be uniquely bicoloured in the necessary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') When constructing  a topological map from a bicoloured ribbon graph, the bicolouring of the boundaries of the  ribbon graph extends naturally to a face bicolouring of the resulting topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the combinatorial setting of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, the necessary bicolouring constraint is  enforced by requiring that EQ admit a partitioning EQ = Er ∪ Eb into sets Er ≃ EQ/τ0 and  Eb = EQ \\ Er of red and black quarter-edges, respectively, such that τ0(Ex) = EQ \\ Ex and  τ1(Ex) = τ2(Ex) = Ex for x = r, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The equivalent condition for the oriented combinatorial  maps of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 is that EH = E′  r ∪ E′  b with E′  b = EH \\ E′  r and τe(E′  x) = τv(E′  x) = EH \\ E′  x,  where we again take x = r, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, writing Er = {1′, 2, 3′, 4} shows that the topological map  of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12 can be bicoloured in a way that relates it to the first ribbon graph displayed  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14, while it can be checked that the oriented combinatorial map of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13  cannot be bicoloured in any valid way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16: Pictured is the bicoloured 2-rooted topological map corresponding to the ribbon  graph of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It has two black (depicted as dark grey) and red faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since it is  embedded in the sphere, it is orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless, the notches on the blue root-markings  indicate that the local orientations of the vertices are inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The topological maps discussed above can be simplified in an interesting way by trans-  forming them into topological hypermaps (whose definition we recall from the end of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In the LUE case, this is done by contracting red faces to red vertices while identifying half-  edges as follows: Traverse clockwise around each vertex, starting at the rooted half-edge, and  identify the rooted half-edge with the second visited half-edge (retaining the root-marking),  the third visited half-edge with the fourth, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This procedure results in a topological  181  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  hypermap consisting of red and black vertices that are connected by (possibly rooted) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The number of faces and red vertices of the resulting topological hypermap is respectively  equal to the number of black and red faces of the original topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, the  act of contracting faces to vertices does not affect the genus of the relevant surface, so the  topological hypermap obtained through the above procedure has the same genus as the  bicoloured topological map (and ribbon graph) that it corresponds to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we may rewrite  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) in terms of topological hypermaps by interpreting the coefficient  #{Γ ∈ Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | g(Γ) = l/2 and Vr(Γ) = p}  as the number of genus l/2 n-rooted topological hypermaps with p red vertices and n  black vertices of valency k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn (it is straightforward to see that the above construction  is invertible and thus bijective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In terms of combinatorial hypermaps, this is equivalent  to counting the number of triples (EH, τe, τv) such that EH has k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' + kn edges, τe has p  cycles, τv has n cycles of length k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn, and τ−1  v τ−1  e  has k1 + · · · + kn + 2 − n − p − l cycles  (this ensures that the number of faces is consistent with the genus equalling l/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17: The topological hypermap illustrated on the right is obtained from the topo-  logical map on the left by contracting the red faces to vertices while making the half-edge  identifications 2 ≡ 1, 4 ≡ 3, 2′ ≡ 1′, and 4′ ≡ 3′ — note that we retain the blue root-  markings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The corresponding combinatorial map is (EH, τe, τv) with EH = {1, 3, 1′, 3′},  τe = (1, 3′, 1′)(3), and τv = (1, 3)(1′, 3′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Observe that τ−1  v τ−1  e  = (1, 3′, 3)(1′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The procedure described above is also valid in the LOE case, but one needs to decorate  the resulting locally orientable topological hypermaps in an appropriate manner to ensure  that the mapping is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To understand this requirement, let us first recall that each  half-edge of a topological map is either of type GT or G, as discussed below equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  182  3  3  2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', in the left image of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17, the half-edges 1, 3, 1′, 3′ are of type GT, while 2, 4, 2′, 4′  are of type G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the case of LUE topological maps, traversing the boundary of a red face  shows that such a boundary is a chain of half-edges with no two consecutive half-edges being  of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is not the case when dealing with LOE topological maps since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the  boundary of the large red face in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 has two rooted half-edges (automatically of type  GT) meeting each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To account for this subtlety, we assign ±1 ‘twist’ labellings (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the  discussion below Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) to the edges of our locally orientable topological hypermaps  in such a way that interchanging the type GT ↔ G of map half-edges whenever they are  represented by twisted hypermap edges (labelled −1) results in topological maps whose red  faces are bounded by a chain of half-edges of alternating type, as seen in the LUE case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note  that the edges labelled −1 are truly twisted in the sense that the faces glued to such edges  must switch sides somewhere along them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18: The left image is the result of gluing a white open disk to a (−1-labelled)  twisted edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To make this possible, we have folded a portion of the disk over the grey  length of the edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the green (blue) lines are attached to the boundary of the white disk at  points that are relatively close to each other and one should be able to imagine a continuum  of similar lines in between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Folding the disk over the grey length of the edge results  in a peak (thin black line) and trough (dotted orange line), which are better visualised  through the depicted cross sections — the first (second) cross section refers to the leftmost  (rightmost) green and blue lines in the illustration on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this case, flattening the  peak and trough (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', untwisting the edge) shows that we simply have a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The alert reader will have noticed that there are multiple ways of assigning twists to  the edges of a topological hypermap so that it correctly corresponds to a given locally  orientable, bicoloured topological map (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', changing the −1 label in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18 to +1 is  183  >Z  X<CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  inconsequential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that each such topological map corresponds to an equivalence  class of ±1-labelled topological hypermaps, where two hypermaps are said to be equivalent  if one can be obtained from the other by using local surgery to sequentially reverse the local  orientations of some vertices while inverting the signs of all edges connected to said vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19: On the top left, we have an LOE ribbon graph with two twisted ribbons and  a blue root-marking on its top polygon edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Gluing a red (dark grey) face to the red  (black) boundary, then shrinking the ribbons to edges and the white square to a black vertex  results in the topological map on the top right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the green ⊗ symbol denotes a cross-cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Contracting the red face to a vertex while identifying half-edges in the necessary manner  then results in either of the two equivalent topological hypermaps shown in the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In comparing Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, one observes the well known fact that gluing  an open disk to a twisted interval (where the twist can be undone without changing its  topology) results in a sphere, while doing the same to a loop formed by one twisted and  untwisted edge (essentially a M¨obius strip) results in a real projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us now give a reformulation of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14 in terms of hypermaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N and define Tk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to be the set of orientable n-rooted  topological hypermaps with k1 + · · · + kn edges, any number of red vertices, and n black vertices of  valency k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn, with each black vertex having one root edge incident to it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' to fix convention, assign  184  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  all black vertices a clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Next, let T∗  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the set of ±1-labelled locally orientable  topological hypermaps generated by taking elements of Tk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, deleting their faces, labelling each  of their edges with either ±1, and then gluing new faces to the boundaries in a manner consistent  with that shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, let ˜T∗  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = T∗  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn/ ∼, where ∼ denotes the equivalence  relation described immediately above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, we have that  c(LOE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  k1+···+kn+1−n  ∑  l=0  � N  2  �k1+···+kn  N2−n−l  ×  k1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='+kn+1−n  ∑  p=1  (ˆa + 1)p #{Ω ∈ ˜T∗  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | ˜g(Ω) = l and Vr(Ω) = p},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  where ˜g(Ω) is the Euler genus of Ω and Vr(Ω) is the number of red vertices of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As we move on to the next subsection, let us remark that the ±1-labelled topological  hypermaps of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15 have combinatorial analogues that can be defined by extending  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 to allow for hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Without delving into the details, for the sake of brevity,  let us simply state that the key ideas are to remove the third condition of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and  to interpret the orbits of τ0, ⟨τ0, τ1⟩, and ⟨τ0, τ2⟩ as edges, red (white in the context of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  vertices or hyperedges, and black vertices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Moments of the Hermitised and antisymmetrised matrix products  In this subsection, we amalgamate concepts from the previous two subsections in order to  derive ribbon graph interpretations for the mixed moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) and cumulants (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  of the Hermitised and antisymmetrised matrix product ensembles introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We first give some basic results for general products of (possibly) rectangular matrices before  highlighting some interesting simplifications in terms of topological hypermaps for the  simplest non-trivial products (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the Hermitised and antisymmetrised Laguerre ensembles  corresponding to taking m = 1 and ν1 = ν0 = 0 in Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Ribbon graphs for the mixed moments of the Hermitised matrix product ensembles  Following Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10, let m, N0 ∈ N, fix N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm ∈ N such that for 1 ⩽ i ⩽ m,  Ni ⩾ N0, and write N := Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) Hermitised matrix product  185  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  ensemble is represented by the N × N product  Hm = G†  m · · · G†  1HG1 · · · Gm,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45)  where H is drawn from the N0 × N0 GUE and for 1 ⩽ i ⩽ m, each Gi is drawn independently  from the Ni−1 × Ni complex Ginibre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting ⟨ · ⟩ now, and for the remainder of  this subsection, denote averages with respect to the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(G)(H)  ���  N�→N0  m  ∏  i=1  P(Gin)(Gi)  ���  (M,N)�→(Ni−1,Ni),  we give a preliminary characterisation of the mixed moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) of the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N)  Hermitised matrix product ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Fix m, n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N and let Hm be as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For 1 ⩽ s ⩽ n and  1 ⩽ t ⩽ ks − 1, set i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  ks  := i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  1  and i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  t  := i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, we have that  m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  �  n  ∏  a=1  ka  ∏  b=1  m+1  ∏  c=1  N|c−1|  ∑  i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±c)  b  =1  �  �  �  n  ∏  s=1  ks  ∏  t=1  Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  �  ×  m  ∏  u=1  �  n  ∏  s=1  ks  ∏  t=1  (G†  u)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u)  t  (Gu)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u−1)  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Taking equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) as the definition of m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and inserting the expansion  Tr Hksm =  N  ∑  i(s)  1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',i(s)  ks =1  (Hm)i(s)  1 i(s)  2 (Hm)i(s)  2 i(s)  3 · · · (Hm)i(s)  ks i(s)  1  shows that  m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  n  ∏  s=1  Tr Hksm  �  =  �  �  n  ∏  a=1  ka  ∏  b=1  N  ∑  i(a)  b =1  �  �  �  n  ∏  s=1  ks  ∏  t=1  (Hm)i(s)  t i(s)  t+1  �  ,  where we have set i(s)  ks+1 := i(s)  1  for all 1 ⩽ s ⩽ n — cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Making the  replacement (Hm)i(s)  t i(s)  t+1 �→ (Hm)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t+1  = (Hm)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  t  and using the definition  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) of Hm to write  (Hm)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  t  = (G†  m)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m)  t  (G†  m−1)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m−1)  t  · ·  · · (G†  1)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  (G1)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−2)  t  · · (Gm)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  t  186  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  then shows that  m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  �  n  ∏  a=1  ka  ∏  b=1  m+1  ∏  c=1  N|c−1|  ∑  i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±c)  b  =1  �  �  �  n  ∏  s=1  ks  ∏  t=1  (G†  m)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m)  t  (G†  m−1)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m−1)  t  · ·  · · (G†  1)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  (G1)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−2)  t  · · (Gm)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As the random matrices H, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Gm within equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) are assumed to be (mutually)  independent, the average in the above can be split into the product of averages shown on  the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46), thereby proving the latter formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As shown in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, each average on the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  can be simplified using the Isserlis–Wick theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, the average  �  n  ∏  s=1  ks  ∏  t=1  Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  �  is identically zero if it contains an odd number of entries, so m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanishes if k1 + · · · + kn  is an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, when k1 + · · · + kn is an even integer, we are led  to the following ribbon graph construction: For each s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n, draw a ks(2m + 1)-gon  and label its vertices in clockwise order in such a way that the first 2m + 1 vertices are  labelled i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  1  , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m)  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−2)  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m)  1  , the next 2m + 1 vertices are labelled  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  2  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  2  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m)  2  , and so on, with the last 2m + 1 vertices being labelled  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  ks  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  ks  , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  ks  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m)  ks  — for 1 ⩽ t ⩽ ks and 1 ⩽ u ⩽ m, edges of the form  (i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  → i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  ) represent Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  , while (i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u+1)  t  → i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u)  t  ) and (i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u)  t  → i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u−1)  t  ) represent  (G†  u)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u)  t  and (Gu)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u−1)  t  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Applying the Isserlis–Wick theorem to the averages on the right-hand side of equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) results in a product of covariances that are exactly of the type seen in the GUE and  LUE cases and can thus be represented by untwisted ribbons connecting polygon edges  representing entries from a matrix and its adjoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' that is, for 1 ⩽ u ⩽ m, if one end of a  ribbon is connected to an edge representing an entry of G†  u (H), then its other end must be  connected to an edge representing an entry of Gu (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This latter constraint can be satisfied  by assigning, for each u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m + 1, a unique colour φu to all polygon vertices labelled  by indices of the form i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±u)  t  , with 1 ⩽ s ⩽ n and 1 ⩽ t ⩽ ks, and then only allowing ribbon  graphs whose boundaries have well-defined colours inherited from the vertices they pass  through;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  187  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20: In computing m(H2)  2,1,1 = ⟨Tr(H2  2) Tr(H2)2⟩, one first draws a decagon, represent-  ing Tr(H2  2), and two pentagons, each representing a factor of Tr(H2), with their vertices  coloured as shown (the colours φ1, φ2, φ3 are respectively red, blue, and green) — we do not  display the vertex labellings, but mention that the top green vertex of each polygon, moving  left to right, is respectively labelled i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  1  , i(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  1  , i(3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  1  , while the other vertex labels follow  from the prescription given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the next step, one simply lists out all possible ways  of pairwise identifying the polygon edges using untwisted ribbons whose sides inherit  well-defined colours from the vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the displayed ribbon graph illustrates one such  pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For visual clarity, and without any mathematical meaning, we have assigned the  colours light blue, purple, and grey to the ribbons with respectively two red sides, a red  and blue side, and a blue and green side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These ribbons represent pairings of the form  ⟨HijHji⟩, ⟨(G†  1)ij(G1)ji⟩, and ⟨(G†  2)ij(G2)ji⟩ (for suitable i, j), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Extending arguments given in the previous two subsections (in particular, the ideas  underlying Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) leads to an elegant formula for the mixed moments (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  and cumulants (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) of the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) Hermitised matrix product ensemble in  terms of ribbon graphs, in analogy with equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Fix m, n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N and let Hm be as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If k1 + · · · + kn is odd,  we have that  m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, constraining ourselves to the case that k1 + · · · + kn is even, define P(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to be the set of  (m + 1)-coloured orientable 1  2(2m + 1)(k1 + · · · + kn)-ribbon graphs that can be constructed through  188  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  the procedure described above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20 and let P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn ⊆ P(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the subset of connected such  ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2− 1  2 (k1+···+kn)  ∑  Γ∈P(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  NV0(Γ)  0  NV1(Γ)  1  · · NVm(Γ)  m  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2− 1  2 (k1+···+kn)  ∑  Γ∈P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  NV0(Γ)  0  NV1(Γ)  1  · · NVm(Γ)  m  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48)  where, for 0 ⩽ u ⩽ m, Vu(Γ) is the number of boundaries of Γ that are of the colour φu+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned above, applying the Isserlis–Wick theorem to the averages in the right-  hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) shows that  �  n  ∏  s=1  ks  ∏  t=1  Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  �  vanishes if k1 + · · · + kn is odd and is otherwise given by a sum over Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (defined above  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)) of a product of indicator functions identifying the summation indices  i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, the averages  �  n  ∏  s=1  ks  ∏  t=1  (G†  u)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u)  t  (Gu)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u−1)  t  �  are given by a similar sum, but over Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (recall Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 0 when  k1 + · · · + kn is odd and, otherwise, the canonical bijection P(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn ≃ Pk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn × (Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn)m  allows us to write the product of averages in the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) as a  sum over P(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn of a product of indicator functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Interchanging the order of summation and summing over all indices i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='c)  b  then produces  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47) — the product of indicator functions in the summand is such that the  number of distinct indices of the form i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±c)  b  , with 1 ⩽ a ⩽ n and 1 ⩽ b ⩽ ka free, but  1 ⩽ c ⩽ m + 1 fixed, is equal to the number of boundaries of colour φc in the associated  ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, we know from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 that c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 0 when k1 + · · · + kn is odd, while for  k1 + · · · + kn even, the discussion above said remark implies equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The ribbon graph of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20 has two red (φ1) boundaries, two blue (φ2)  boundaries, and three green (φ3) boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, by the above lemma, it contributes a  value of N2  0 N2  1 N3  2 to both m(H2)  2,1,1 and c(H2)  2,1,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  189  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  In the context of Chapter 4, our interest lies in the mixed cumulants, as opposed to the  mixed moments, since the former have (under appropriate conditions) genus expansions  with convergence properties that allow their generating functions (after correct scaling) to  have large N0 expansions of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we now give the analogue of the genus  expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) pertaining to the global scaled Hermitised matrix product ensembles, as  is relevant to the development of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let the parameters m, N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm = N ∈ N and matrices H, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Gm be as  in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10, equivalently equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), and let us require that the parameters νi := Ni − N0  be such that for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m, there exists a non-negative constant ˆνi := νi/N0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Furthermore, introduce the scaled GUE matrix ˜H := (N0/2)−1/2H and, for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m, the  scaled complex Ginibre matrices ˜Gi := (NiNi−1)−1/4Gi (following [58]) so that  ˜Hm := ˜G†  m · · · ˜G†  1 ˜H ˜G1 · · · ˜Gm =  N−m−1/2  0  ( ˆν1 + 1) · · · ( ˆνm−1 + 1)  �  2  ˆνm + 1Hm  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49)  represents the global scaled (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) Hermitised matrix product ensemble (refining  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In this regime, and for n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N such that k1 + · · · + kn is even, the mixed  cumulants of this ensemble have the genus expansion  ˜c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  ( ˆν1 + 1) · · · ( ˆνm−1 + 1)  �  ˆνm + 1  �−k1−···−kn N2−n  0  ×  1  2 (2m+1)(k1+···+kn)+1−n  ∑  l=0  1  Nl  0  k1+···+kn  ∑  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',pm=1  ( ˆν1 + 1)p1 · · · ( ˆνm + 1)pm  × #{Γ ∈ P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | g(Γ) = l/2 and Vi(Γ) = pi for each 1 ⩽ i ⩽ m},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50)  where the set P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and the functions V1(Γ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Vm(Γ) are as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 above, and we recall  that g(Γ) denotes the genus of the ribbon graph Γ, while #S denotes the cardinality of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting Ni = ( ˆνi + 1)N0 (1 ⩽ i ⩽ m) into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48) shows that  c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2− 1  2 (k1+···+kn)  1  2 (2m+1)(k1+···+kn)+2−n  ∑  V=1  NV  0  k1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='+kn  ∑  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',pm=1  ( ˆν1 + 1)p1 · · · ( ˆνm + 1)pm  × #{Γ ∈ P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn |  m  ∑  i=0  Vi(Γ) = V and Vi(Γ) = pi for each 1 ⩽ i ⩽ m},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)  where the upper terminal of the sum over V, the total number of boundaries of the ribbon  graphs Γ, is obtained through a similar argument to that given in the paragraph containing  190  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19): In the present setting, the polygonised surface obtained by identifying  polygon edges of a given Γ ∈ P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn by collapsing the ribbons therein to their ends has F = n  faces (number of polygons), E = 1  2(2m + 1)(k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' + kn) distinct edges (half the number  of polygon edges that are now pairwise identified), V distinct vertices, and non-negative  integer genus g, so that consideration of the Euler characteristic χ = F − E + V = 2 − 2g  reveals that  V = 1  2(2m + 1)(k1 + · · · + kn) + 2 − 2g − n ⩽ 1  2(2m + 1)(k1 + · · · + kn) + 2 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Using this equation, with l := 2g, to change variables in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) then yields the  result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50), bar a factor of [( ˆν1 + 1) · · · ( ˆνm−1 + 1)  �  ( ˆνm + 1)/2Nm+1/2  0  ]−k1−···−kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  factor, which is the (k1 + · · · + kn)th power of the scaling factor in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49), appears  as the ratio of scaled and unscaled mixed moments  ˜m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  =  �  ∏n  s=1 Tr ˜Hksm  �  �  ∏n  s=1 Tr Hksm  �  =⇒ m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  ( ˆν1 + 1) · · · ( ˆνm−1 + 1)  �  ( ˆνm + 1)/2Nm+1/2  0  �k1+···+kn  ˜m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  inserting this expression into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20) finally shows that said factor is also equal to  the ratio ˜c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn/c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Before moving forward, let us make two remarks regarding Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The first  remark is that, since P(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn consists of orientable ribbon graphs whose genera are non-  negative integers, Nn−2  0  ˜c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is an even polynomial in N−1  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, upon setting νi = ˆνiN0  with ˆνi = O(1) (1 ⩽ i ⩽ m) and writing the generating functions ˜W(Hm)  n  of the scaled  cumulants ˜c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn in the form given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, one sees that the related expansion  coefficients W(Hm),l  n  are identically zero for odd values of l, in line with what is known in  the GUE and LUE cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our second remark relates to the choice of setting νi = ˆνiN0 as  just discussed, rather than νi = ˆνiN0 + δi with δi = O(1), as in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This choice,  as with the choice of setting a = ˆaN with ˆa = O(1) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, is made so that equations  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61) upcoming convey their combinatorial content in the  most elegant way possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless, the relevant proofs follow through in the more  general setting where not all δi are identically zero, albeit the expressions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61) cease to hold true past leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  191  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  Ribbon graphs for the mixed moments of the antisymmetrised matrix product ensembles  Let us now take N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm to be positive even integers with N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm ⩾ N0 and otherwise  retain our specifications of the parameters m, N = Nm ∈ N and notation ⟨ · ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover,  for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m, let Gi now be drawn independently from the Ni−1 × Ni real Ginibre  ensemble and recall that JN0 denotes the N0 × N0 elementary antisymmetric matrix defined  in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) — for present purposes, it suffices to observe that JT  N0 = −JN0 and  J2  N0 = −JT  N0 JN0 = −IN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' According to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11, the N × N product  Jm = GT  m · · · GT  1 JN0G1 · · · Gm  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  represents the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) antisymmetrised matrix product ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We begin with  the analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 for the mixed moments of this ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Fix m, n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N and let Jm be as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The mixed moment  m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanishes if any of the k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Otherwise, we have that  m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  �  n  ∏  a=1  ka  ∏  b=1  m+1  ∏  c=1  N|c−1|  ∑  i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±c)  b  =1  �  �  �  n  ∏  s=1  ks  ∏  t=1  (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  �  ×  m  ∏  u=1  �  n  ∏  s=1  ks  ∏  t=1  (GT  u )i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u+1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='u)  t  (Gu)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−u−1)  t  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53)  where we have retained the identifications (introduced in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  ks  := i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  1  and  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−m−1)  t  := i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='m+1)  t+1  for 1 ⩽ s ⩽ n and 1 ⩽ t ⩽ ks − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If ks is odd for some 1 ⩽ s ⩽ n, then the antisymmetric nature of Jm means that  Tr J ks  m vanishes and, consequently, so too does m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When all of the  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn are even, the proof is as for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Replicating the construction preceding Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20 with equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) in mind shows  that we are interested in locally orientable ribbon graphs that are built in a similar fashion  to those in the Hermitised case: The computation of m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn requires one to build ribbon  graphs from n polygons with respectively k1(2m + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn(2m + 1) sides and with vertices  labelled and coloured in the exact same manner as in the case of m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, but the ribbons are  now allowed to be M¨obius half-twisted and there are no longer any ribbons connected to  192  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  the polygon edges of the form (i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  → i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  ), which now represent (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  instead  of Hi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Of course, we still require that the sides of the ribbons connect vertices of  like colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since the matrices G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Gm are independent from each other, the collection  of locally orientable ribbon graphs just described, which we denote ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, is in bijection  with ( ˜Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn)m, where ˜Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is the extension of ˜Sc  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) allowing for  disconnected ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In other words, the ribbon graphs of ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn can be constructed  by appropriately combining m distinct LOE ribbon graphs drawn from ˜Sk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In that vein,  it is helpful to compare the ribbon graphs of ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to those obtained by deleting the edges  representing entries of JN0 while identifying, for 1 ⩽ s ⩽ n and 1 ⩽ t ⩽ ks, the vertices  labelled i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  to those labelled i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', replacing each instance of (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  in equation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) with χi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  =i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  , so that this equation now computes the mixed moments m(rWm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn of  the real Wishart product matrix Wm = GT  m · · · GT  1 G1 · · · Gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21: The ribbon graph on the left contributes to the computation of m(J2)  2  and  belongs to ˜S(2)  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Deleting the orange edges representing entries of JN0 and identifying their  ends results in the middle ribbon graph, which we think of as belonging to ( ˜S2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  latter ribbon graph is a combination of the two ribbon graphs on the right (cut and glue  along the dotted blue lines), each of which are drawn from ˜S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Applying the Isserlis–Wick theorem to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) in the same fashion as in the  proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 produces the following analogue of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Fixing m, n ∈ N, let Jm be as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52), and take k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let  ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the set of (m + 1)-coloured locally orientable m(k1 + · · · + kn)-ribbon graphs that can  be constructed in the manner described above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21 and define fΓ({i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  }1⩽s⩽n, 1⩽t⩽ks) to be  193  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  the product of indicator functions that encodes the identifications of the vertices of colour φ1 that are  represented by the ribbon data of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, we have that  m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2−m(k1+···+kn)  ∑  Γ∈ ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  NV1(Γ)  1  · · NVm(Γ)  m  �  �  n  ∏  a=1  ka  ∏  b=1  N0  ∑  i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  b  =1  �  �  �  n  ∏  s=1  ks  ∏  t=1  (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  �  × fΓ  �  {i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  }1⩽s⩽n, 1⩽t⩽ks  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  where V1(Γ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Vm(Γ) are as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting Γ be the left illustration of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21 and following the red sides of  the purple ribbons therein shows that  fΓ  �  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  , i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  , i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  , i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  2  �  = χi(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  =i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  χi(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  =i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='55)  The product fΓ({i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  }1⩽s⩽n, 1⩽t⩽ks) is such that if one replaces (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  by χi(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  =i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54), as discussed above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21, said equation reduces to an expression  for the mixed moments of the corresponding real Wishart product ensemble:  m(rWm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2−m(k1+···+kn)  ∑  Γ∈ ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  NV0(Γ)  0  · · NVm(Γ)  m  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56)  where V0(Γ) is the number of boundaries of the ribbon graph Γ of colour φ1 after all of the  edges representing entries of JN0 have been given this colour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equivalently, it is the number  of such boundaries of the ribbon graph obtained by collapsing said edges of Γ to their ends  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the middle illustration of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21, which has φ1 being red and thus V0(Γ) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence,  one can think of the contribution of each ribbon graph Γ ∈ ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to the value of m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  as being equal to a perturbation of the value 2−m(k1+···+kn)NV0(Γ)  0  · · NVm(Γ)  m  contributed by  Γ to m(rWm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn via equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It turns out that the correct way to perturb this value is  to replace the factor NV0(Γ)  0  by an appropriate product of traces of monomials in JN0 and  JT  N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To see this, first recall that in the real Wishart product case, the factor NV0(Γ)  0  arises  from assigning a weight N0 to each distinct boundary, equivalently polygon vertex, of colour  φ1 in Γ and that traversing these boundaries tells us how to identify vertex labels so as  to arrive at the set of distinct labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Upon reintroducing the polygon edges representing  entries of JN0, traversing these same boundaries instead tells us how to identify indices of  these matrix entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, to obtain the weight assigned to the ‘first’ boundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', that  194  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  containing the vertex labelled i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  , let ω1 = (JN0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  and then follow the ribbon border  of colour φ1 connected to the vertex labelled i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  to see which vertex it is identified to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If  this latter vertex has label of the form i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  , identify this label with i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  and rewrite ω1  as the ordered string (JN0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  (JN0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, if said vertex has label  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  , rewrite ω1 as (JN0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  (JT  N0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Next, follow the ribbon border connected to  the vertex labelled i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  and likewise extend ω1 by concatenating it with a term of the form  (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  or (JT  N0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  , depending on if the ribbon data identifies i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  with i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  or  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Continue on in this manner until the boundary has been fully traversed  (that is, the first index of the first matrix entry of ω1 matches the second index of the last  matrix entry of ω1) and then repeat this exercise for all of the remaining boundaries of colour  φ1 so that we end up with a list of words ω1, ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ωV0(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With this list of words in hand,  we may now give a refinement of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For each word ωj obtained via the procedure described above, let |ωj| denote the length  of ωj and let sgn(ωj) := (−1)#JT, where #JT is the number of entries of JT  N0 in ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the setting of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7, we then have that  m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 2−m(k1+···+kn)  ∑  Γ∈ ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  NV0(Γ)  0  · · NVm(Γ)  m  V0(Γ)  ∏  j=1  sgn(ωj) Re  �  i|ωj|�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First, observe that suitably ordering the terms in the summand of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  and rewriting some factors (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  as (JT  N0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  shows that  �  n  ∏  s=1  ks  ∏  t=1  (JN0)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  �  × fΓ  �  {i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  }1⩽s⩽n, 1⩽t⩽ks  �  = ω1 · · · ωV0(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Hence, summing this expression over all indices i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  results in a weight of  �  �  n  ∏  a=1  ka  ∏  b=1  N0  ∑  i(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  b  =1  �  � ω1 · · · ωV0(Γ) = Tr  �  q(1)  Γ (JN0, JT  N0)  �  · · Tr  �  q(V0(Γ))  Γ  (JN0, JT  N0)  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58)  where q(j)  Γ  (1 ⩽ j ⩽ V0(Γ)) is the monomial obtained from the string ωj by erasing matrix  indices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the string (JN0)ab(JT  N0)bc(JN0)ca induces the monomial JN0 JT  N0 JN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note also that  Tr  �  q(j)  Γ (JN0, JT  N0)  �  = Tr  �  q(j)  Γ (JN0, −JN0)  �  = Tr  �  q(j)  Γ (1, −1)J  deg q(j)  Γ  N0  �  = q(j)  Γ (1, −1) Tr  �  J  deg q(j)  Γ  N0  �  ,  195  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  which simplifies to either zero, N0, or −N0 (we have abused notation to define q(j)  Γ on R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since for k ∈ N, J2k  N0 = (−IN0)k = (−1)kN0 and Tr J2k+1  N0  = 0, we see that  q(j)  Γ (1, −1) = sgn(ωj),  Tr  �  J  deg q(j)  Γ  N0  �  = N0 Re  �  i|ωj|�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, the weight of the jth boundary is  Tr  �  q(j)  Γ (JN0, JT  N0)  �  = N0 sgn(ωj) Re  �  i|ωj|�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='59)  Substituting this into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='58) and the result of doing so into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us briefly revisit the ribbon graph depicted on the left in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  weight corresponding to the blue and green vertices is N1N2  2 since there are one blue and  two green boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Traversing the red line starting at the vertex corresponding to the  label i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  shows us that i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  ≡ i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  2  , in keeping with equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='55), so we write  ω1 = (JN0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  (JT  N0)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In a larger ribbon graph, we would then look to extend ω1  via concatenations, but we see here that i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  ≡ i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  , so we are done (indeed, the red and  orange boundary has been fully traversed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We see that ω1 contains an entry of JN0 followed  by an entry of JT  N0, so we have that sgn(ω1) = (−1)1 = −1 and |ω1| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the weight  of the single red and orange boundary is (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='59)  Tr(JN0 JT  N0) = −N0 Re  �  i2� = N0  and, hence, our ribbon graph contributes a value of N0N1N2  2/16 to the computation of m(J2)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One way to think about the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57) is that it is simply the sum (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56)  with the summand having each factor of N0 replaced either zero, N0, or −N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is possible  to apply the arguments given above to the case of the Hermitised product ensembles so  that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47) can likewise be interpreted as being given by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56) with  the sum being constrained to be over only the orientable ribbon graphs in ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and with  the factor of NV0(Γ)  0  in the summand replaced by the GUE mixed moment m(GUE)  k′  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',k′  V0(Γ)|N�→N0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18), where k′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k′  V0(Γ) ∈ N are such that the ith boundary of colour φ1 seen in Γ passes  through k′  i vertices of that colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As an example, we would weight the middle image of  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21 by m(GUE)  2  since there is one red boundary passing through two red vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  196  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Returning now to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8, let us give the analogue of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 in the case of  the antisymmetrised matrix product ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let the parameters m, N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm = N ∈ N and matrices JN0, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Gm be as  in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11, equivalently equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52), and let us require that the parameters νi := Ni−N0  2  be such that for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m, there exists a non-negative constant ˆνi := νi/N0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Furthermore, introduce the scaled real Ginibre matrices ˜Gi := ( NiNi−1  4  )−1/4Gi (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m) so that  ˜Jm := ˜GT  m · · · ˜GT  1 JN0 ˜G1 · · · ˜Gm =  2mN−m  0  (2ˆν1 + 1) · · · (2ˆνm−1 + 1)√2ˆνm + 1Jm  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='60)  represents the global scaled (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) antisymmetrised matrix product ensemble  (again refining Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, for n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N, the mixed cumulants of this ensemble  vanish if any of the k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn are odd and are otherwise given by the genus expansion  ˜c(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  (2ˆν1 + 1) · · · (2ˆνm−1 + 1)  �  2ˆνm + 1  �−k1−···−kn N2−n  0  ×  m(k1+···+kn)+1−n  ∑  l=0  1  Nl  0  k1+···+kn  ∑  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',pm=1  (2ˆν1 + 1)p1 · · · (2ˆνm + 1)pm  ×  ∑  Γ∈ ˜C(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  p0  ∏  j=1  sgn(ωj) Re  �  i|ωj|�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61)  where the strings ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ωp0 are as introduced above Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 and we define  ˜C(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn := {Γ ∈ ˜S(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | ˜g(Γ) = l and Vi(Γ) = pi for each 0 ⩽ i ⩽ m},  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='62)  with V0(Γ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Vm(Γ) as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 (having assigned the colour φ1 to the polygon edges rep-  resenting entries of JN0) and with ˜S(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn being the subset of connected ribbon graphs in ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  (defined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' recall from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that ˜g(Γ) denotes the Euler genus of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Observe that m(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is a weighted count of ribbon graphs with the necessary property  for it to be amenable to the inductive argument given in the paragraph above Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4:  the weight of a disconnected ribbon graph is the product of the weights of its connected  components (note that the interpretation of m(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn given in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 does not have this  property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, by the aforementioned argument, the mixed cumulants of the antisym-  metrised matrix product ensembles are given by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57), but with ˜S(m)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn replaced  by ˜S(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The fact that the cumulants vanish whenever any of the k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn are odd follows  197  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  from the same reasoning as in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4, while the remainder of this proposition is proved  in the same manner as Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 as our starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since the third line of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61) is manifestly an integer, it is immediate that  N2−n  0  ˜c(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is a polynomial in N−1  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, unlike what was seen in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16, this  is not an even polynomial, since we now allow for non-orientable ribbon graphs with odd  Euler genus (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', terms corresponding to odd values of l now have non-zero contribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  this is exactly what was seen in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 when comparing the GUE/LUE to the  GOE/LOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another point of contrast is that the mixed cumulants ˜c(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn can take negative  values, unlike ˜c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (and, indeed, all other mixed cumulants discussed in this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We recognise that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 is a little less intuitive than Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 due to the  awkward manner in which the words ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ωp0 must be constructed for each Γ ∈ ˜C(m),c  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It turns out that the third line of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61) can be simplified further, but we do not  present this simplification here, opting instead to discuss it in the m = 1, ν1 = ν0 = 0 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Topological hypermaps in the setting with m = 1 and ν1 = ν0 = 0  We now use the ideas presented at the ends of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to derive topological  hypermap formulations of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To further simplify our results, we  restrict ourselves to the m = 1 case concerning products of square matrices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', N1 = N0 = N  and ν1 = ν0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our arguments are similar to those given briefly in the case of the m = 2,  (N, N, N) complex Wishart product ensemble in [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It is pertinent that we point out that  the upcoming derivations do not have any natural extensions to the m > 1 cases, but that  such cases can be treated combinatorially by introducing the theory of constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do  not discuss constellations here and instead refer the reader to the textbook [220].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us first consider the (N, N) Hermitised Laguerre ensemble represented by the N × N  matrix H1 = G†HG with G drawn from the N × N complex Ginibre ensemble and H  drawn from the N × N GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From the discussion preceding Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20, we know that  the mixed cumulants c(Hm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) of this ensemble can be computed by enumerating  connected, orientable ribbon graphs constructed in the following manner: Draw n polygons  with respectively 3k1, 3k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , 3kn sides, pick one vertex of each polygon to be the ‘first’ or  rooted vertex of that polygon and then, moving clockwise around the polygon while starting  198  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  at the first vertex, colour each sequence of three vertices red, yellow, and yellow, in that  order (taking φ1, φ2 to be yellow and red, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Next, connect polygon edges with  untwisted ribbons whose sides identify vertices of like colour in a way that results in a  connected ribbon graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For visual clarity, we also colour the ribbons connecting edges  representing entries of G† (H) to those representing G (H) black (purple);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22  below for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These ribbon graphs can immediately be reformulated as n-rooted  topological maps by gluing yellow (red) faces to the yellow (red) boundaries, shrinking  the polygons to vertices, thinning the ribbons to edges connecting said vertices, and root-  marking the half-edges appearing immediately clockwise to rooted vertices (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To simplify even further to n-rooted topological hypermaps (which we recall were  introduced at the end of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), we contract the red faces to red vertices while identifying  black half-edges in the manner shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17, and then split the purple edges into  two by inserting green vertices in the middle of them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' again, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22: The ribbon graph on the left contributes to the computation of c(H1)  2,1,1 (obtained  by removing grey ribbons from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The blue arrows point to the rooted vertices  and the rooted polygon edges are likewise coloured blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Gluing faces to the boundaries  and shrinking the ribbons and polygons results in the toroidal (genus one) topological map  depicted on the top right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Inserting green vertices in the middle of the purple edges and  shrinking the red faces in the aforedescribed manner then yields the topological hypermap  on the bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The blue edges are actually root-marked black edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We now give a reformulation of the m = 1, ν1 = ν0 = 0 case of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 in terms  199  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  of these particular topological hypermaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N with k1 + · · · + kn even, let Hk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the set of orientable  n-rooted topological hypermaps consisting of k1 + · · · + kn black edges and as many purple edges,  1  2(k1 + · · · + kn) green vertices, any number of red vertices, and n black vertices such that  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the green vertices are bivalent with only purple edges attached to them,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the red vertices only have black edges attached to them,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n, the ith black vertex has ki purple and black edges attached to it in alternating  cyclic order,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n, one black edge incident to the ith black vertex is root-marked and labelled by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Then, appropriately simplifying equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50) shows that the mixed cumulants of the global  scaled (N, N) Hermitised Laguerre ensemble (represented by ˜H1 := ˜G† ˜H ˜G with ˜G := G/  √  N and  ˜H := √  2/NH) is given by  ˜c(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = N2−n  3  2 (k1+···+kn)+1−n  ∑  l=0  1  Nl #{Ω ∈ Hk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn | g(Ω) = l/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63)  Moving on, consider now the (N, N) antisymmetrised Laguerre ensemble represented by  the N × N matrix J1 = GTJNG, where N is an even integer, G is drawn from the N × N real  Ginibre ensemble, and JN is the N × N elementary antisymmetric matrix defned in equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From the discussion leading to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8, we know that the mixed cumulants c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  can be computed by counting LOE ribbon graphs with appropriate weights — the weight of  each ribbon graph is the product of the weights of its boundaries, with black boundaries  being weighted by N and red boundaries being weighted by the trace of a suitable monomial  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='59) in JN and JT  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall that these monomials can be determined by replacing each red  vertex of a given LOE ribbon graph Γ by a new polygon edge representing an entry of JN and  then traversing the red boundaries while constructing the strings ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , ωVr(Γ) described  above Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8, where Vr(Γ) is the number of red boundaries of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Translating this idea  to the setting concerning the associated topological maps, which are obtained by gluing  red (black) faces to the red (black) boundaries and then shrinking polygons and ribbons to  vertices and edges, we need to compute a string ωj for each red face of the topological map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  200  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  Now, recall from the discussion between Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18 that the LOE topolog-  ical maps have half-edges either of type GT or G and further observe that half-edges of  type GT (G) correspond to ribbon ends that make contact with vertices with labels of  the form i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  (i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' here, we remember that we are working with ribbon graphs that  have been obtained by replacing the red vertices of LOE ribbon graphs with polygon  edges representing JN and we have adopted the vertex labelling convention given above  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, an edge of type GT − GT (G − G) signifies an identification of the  form i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  ≡ i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  (i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  ≡ i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  ) so that in the relevant string ωj, we have either of the  substrings (JT  N)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  (JN)i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  or (JT  N)i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  (JN)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  ((JN)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  (JT  N)i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  or  (JN)i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  (JT  N)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  ) appearing — edges of type GT − GT and G − G correspond to sub-  strings of the form (JN)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  (JN)i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  or (JT  N)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  (JT  N)i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t′  i(s′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, rewriting a  term (JN)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  as (JT  N)i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  t  i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  t  or vice versa in a given string ωj is equivalent to interchang-  ing the types GT ↔ G of the half-edges representing the ribbon ends that interact with the  vertices labelled i(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='±1)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Consequently, the number of JT  N seen in the monomial q(j)  Γ (JN, JT  N)  induced by ωj is equal to the minimum number of times such GT ↔ G interchanges would  need to be applied to the jth red boundary to yield a red boundary made up of a chain of  half-edges of alternating type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, these are the same GT ↔ G interchanges discussed  between Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18: They can be kept track of by transforming our topological  maps into hypermaps through the manner shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and then applying a ±1  labelling to the hypermap edges following the prescription given above Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  the monomial trace Tr q(j)  Γ (JN, JT  N) weighting the jth red map face is equal to Tr J  a+  j  N (JT  N)a−  j ,  where a±  j is equal to the number of edges of type ±1 incident to the red hypermap vertex  corresponding to the jth red map face — consequently, we have that  sgn(ωj) = (−1)a−  j ,  |ωj| = a−  j + a+  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='64)  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Although the above prescription is quite longwinded, it is much easier to  compute the weight of a given topological hypermap than the ribbon graph it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For  example, if we want to compute the contribution of the ribbon graph displayed on the top  left of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19 to the value of c(J1)  2  , we first need to replace the red vertices with diagonal  edges so that the white square becomes a hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The new edge in the top right represents  (JN)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  , while the new edge in the bottom left represents (JN)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  i(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following the  201  CHAPTER 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Characterisations of the Moments and Cumulants  red ribbon border starting at the bottom vertex of the edge representing (JN)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  shows  that i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  ≡ i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  2  and continuing to traverse this red boundary by following the other red  ribbon border confirms that i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  2  ≡ i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we have ω1 = (JN)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  (JT  N)i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='−1)  1  i(1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  1  and  so this boundary is weighted Tr JNJT  N = N sgn(ω1) Re(i|ω1|) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Since we also have one  black boundary, which is automatically weighted N, this ribbon graph contributes a value of  N2/4 to c(J1)  2  in our ν1 = ν0 = 0 setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now, if we look instead at either of the topological  hypermaps given in the bottom of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, we immediately see that the red vertex has  one twisted and one untwisted edge adjacent to it, so it is weighted by N(−1)1Re(i2) = N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  there being one black vertex means that the hypermap is weighted N2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Working with topological hypermaps rather than ribbon graphs allow us to derive the  following simplification of the m = 1, ν1 = ν0 = 0 case of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For n ∈ N and k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ 2N, let ˜Ak1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be the subset of ±1-labelled locally  orientable n-rooted topological hypermaps in the set ˜T∗  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15, whose  vertices all have even valency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, let ˜A ˜g,±  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn ⊆ ˜Ak1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be such that for each Ω ∈ ˜A±  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  Ω has Euler genus ˜g and the product of the signs of the edges of Ω is ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, the mixed cumulants  of the global scaled (N, N) antisymmetrised Laguerre ensemble (represented by ˜J1 := ˜GTJN ˜G with  ˜G = √  2/NG) is given by  ˜c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = (−1)  1  2 (k1+···+kn)N2−n  k1+···+kn+1−n  ∑  l=0  1  Nl  �  # ˜Al,+  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn − # ˜Al,−  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65)  where we recall that #S denotes the number of elements in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Having converted the ribbon graphs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 (with m = 1) into topological  hypermaps in the manner prescribed above, we first note that hypermaps with black vertices  of odd valency have vanishing weight: Recall from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 that the cumulants  c(Jm)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanish if any of the ki are odd — the black vertices of our topological hypermaps  have valency k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Next, note that the jth red vertex (which corresponds to the jth red boundary of the  original ribbon graph) has weight sgn(ωj) Re(i|ωj|) according to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17, which  simplifies to  sgn(ωj) Re(i|ωj|) = (−1)a−  j Re(ia−  j +a+  j )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='66)  202  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Isserlis–Wick Theorem and Ribbon Graphs  by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This weight also vanishes if a−  j + a+  j , the valency of the jth red vertex, is  odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, only hypermaps with all vertices of even valency contribute to the value of the  mixed cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, using equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='66) in the third line of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61) shows that the weight  of a topological hypermap due to the weights of the red vertices is the product  V0(Γ)  ∏  j=1  sgn(ωj) Re(i|ωj|) =  V0(Γ)  ∏  j=1  (−1)a−  j Re(ia−  j +a+  j ),  where, following Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17, V0(Γ) is the number of red boundaries  of Γ and is thus the number of red vertices of the hypermap at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As our weight is  non-vanishing only when a−  j + a+  j is even for all j, we see that  V0(Γ)  ∏  j=1  Re(ia−  j +a+  j ) =  V0(Γ)  ∏  j=1  (−1)  1  2 (a−  j +a+  j ) = (−1)  1  2 (a−  1 +···+a−  V0(Γ)+a+  1 +···+a+  V0(Γ)) = (−1)  1  2 (k1+···+kn),  where the last line follows from the fact that the sum of the valencies a−  j + a+  j of the red  vertices is equal to the total number of edges in the hypermap, which is k1 + · · · + kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  is the source of the factor seen at the front of the right-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the  other hand, since each edge is incident to exactly one red vertex, ∏  V0(Γ)  j=1 (−1)a−  j is simply the  product of the signs of all of the edges present in the hypermap, which leads to the definition  of ˜A±  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting m = 1, ν1 = ν0 = 0 into equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='61) while incorporating the  simplifications just discussed produces the sought formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As we move on to the next chapter, let us mention that, although we have not discussed  them in full generality, many of the arguments leading to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19 can be extended to  the general m ∈ N setting of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Our reason for not presenting these arguments  in this general setting is simply that in the related ribbon graphs, the extra ribbons that  interact with polygon edges representing entries of GT  2 , G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , GT  m, Gm are distracting while  being completely irrelevant to our discussion, as they do not interact with the polygon edges  representing entries of JN0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  203  Chapter 4  Loop Equations for the Matrix Product  Ensembles  In this chapter, we take the first step in exploring the feasibility of studying the Hermitised  and antisymmetrised matrix product ensembles introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 using the loop  equation formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 that for m ∈ N and N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm ∈ N such  that N1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm ⩾ N0, the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) (writing N := Nm) Hermitised matrix product  ensemble is represented by the N × N random matrix product  Hm := G†  m · · · G†  1HG1 · · · Gm,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  where H is drawn from the N0 × N0 GUE and for 1 ⩽ i ⩽ m, each Gi is drawn independently  from the Ni−1 × Ni complex Ginibre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If we further take N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm to be even and  now draw each Gi from the Ni−1 × Ni real Ginibre ensemble, we have by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 that  the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) antisymmetrised matrix product ensemble is represented by  Jm := GT  m · · · GT  1 JN0G1 · · · Gm,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  where JN0 is the elementary antisymmetric matrix defined in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equivalently,  in the notation of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) Hermitised matrix product ensemble  E (Hm) = (S(Hm), P(Hm)) is the set S(Hm) := S(G)��  F=C (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) of N × N complex Hermitian  matrices with the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(Hm)(X) := δ(X − Hm) P(G)(H)  ���  β=2,N�→N0  m  ∏  i=1  P(Gin)(Gi)  ���  β=2,(M,N)�→(Ni−1,Ni),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  204  where δ is the Dirac delta and the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s P(Gin)(G) of the Ginibre ensemble and P(G)(H) of  the Gaussian ensemble are respectively specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, while  the (N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm−1, N) antisymmetrised matrix product ensemble E (Jm) = (S(Jm), P(Jm)) is  the set S(Jm) of N × N antisymmetric real matrices with the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(Jm)(X) := δ(X − Jm)  m  ∏  i=1  P(Gin)(Gi)  ���  β=1,(M,N)�→(Ni−1,Ni),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  where the parameters N0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nm and matrices G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Gm have been appropriately redefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  What we mean by ‘taking the first step’ is that we will be studying the simplest examples of  these ensembles, which correspond to setting m = 1 and N1 = N0 = N — as we will soon  see, taking m to be any greater would result in loop equations too unwieldy to display in the  present setting, while many of the interesting features of such loop equations are already  present in the m = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the language of Definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11, we will thus be  deriving loop equations for the Hermitised and antisymmetrised Laguerre ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  One of our motivations for initiating the aforementioned study is that while the eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s of the Hermitised and antisymmetrised matrix product ensembles have recently  been made explicit in the works [143], [144], as reviewed in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 (technically, we discuss  the non-zero real eigenvalues of the matrix Hm and the positive real eigenvalues of the  matrix iJm), their moments and associated moment generating functions present challenges  relative to the setting of the classical matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, as with our studies of the  classical matrix ensembles presented in the earlier parts of this thesis, we would like to  derive recursive characterisations of, say, the spectral moments mk (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) and resolvents  W1(x) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) of the Hermitised and antisymmetrised matrix product ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' One  might then suggest that we study the ensembles at hand using the methodology laid out  in Chapters 2 and 3 to obtain linear differential equations on the resolvent expansion  coefficients Wl  1(x) and 1-point recursions on the moment expansion coefficients Mk,l defined  implicitly through the expansions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall, though, that the development  of these chapters is based on the Selberg correlation integral theory reviewed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1,  which can only be used to study eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s that are (tractably) expressible in terms  of the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) seen in the Selberg integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is actually the case for the  antisymmetrised Laguerre ensemble, since it is an example of a Laguerre Muttalib–Borodin  ensemble (recall Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11 and Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) and the work [125] relates such Muttalib–  205  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Borodin ensembles to the Selberg integral, so it may therefore be possible to apply the  techniques of Chapters 2 and 3 to the antisymmetrised Laguerre ensemble, but we do not  pursue this line of research since it is quite unlikely that this approach could be extended  effectively to the general m > 1 ensembles E (Jm) and E (Hm), as we currently have no way of  relating the relevant eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s (recall their specifications in terms of the relatively  complicated Meijer G-functions given in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) to the Selberg integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Instead, we  are led to considering the loop equation formalism, which is well known to be applicable to  a large variety of random matrix ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The (final sets of) loop equations derived in this chapter constitute triangular recursive  systems on the coefficients Wl  n of the large N expansions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) = N2−n  ∞  ∑  l=0  Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  Nl  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  of the connected n-point correlators Wn (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29), meaning that they facilitate the systematic,  recursive computation of said coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Although their forming triangular recursive  systems means that these loop equations are not as efficient as corresponding 1-point  recursions (if they exist) for computing the spectral moments m(H1)  k  and m(J1)  k  , they have  the added benefit of producing the coefficients W(H1),l  n  and W(J1),l  n  for n ⩾ 1 and l ⩾ 0,  from which we can extract coefficients of the genus expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65) of the  corresponding mixed cumulants ˜ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn, thereby giving us a method of enumerating the  ribbon graphs and topological hypermaps constructed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Another motivation for deriving loop equations for the Hermitised and antisymmetrised  Laguerre ensembles is that they are of higher order than the equivalent loop equations for the  random matrix ensembles that are more commonly studied in the literature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  forthcoming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, we will see in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 that the loop equations specifying  W0  1(x), also know as the spectral curves [109], for the antisymmetrised (Hermitised) Laguerre  ensembles are third (fourth) order polynomials in W(J1),0  1  (x) (W(H1),0  1  (x)), whereas the  spectral curves of the classical matrix ensembles reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 are quadratic polynomials  in W0  1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Spectral curves of order higher than two have garnered interest in the abstract  setting [110], [45], [46], [268], but there have not been many concrete examples of such  higher order spectral curves and, more broadly, loop equations (here, higher order means  higher than usual) arising from random matrix theory (see, however, the studies on so-called  206  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  multi-matrix models [110]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned above, we produce such examples through the  analysis carried out in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 below, complementing the higher order loop  equations given in the recent work [74] on the (N, N, N) complex Wishart product ensemble  (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that even though the Hermitised and antisymmetrised Laguerre  ensembles are closely related to the complex Wishart product ensemble studied in [74], the  associated loop equations are structurally quite different (bar the similarities one expects  from universality principles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These differences can be intuitively understood from the  discussion of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3: At the level of ribbon graphs and topological hypermaps, all three of  these ensembles are distinct variations of the Laguerre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we introduce the loop equation formalism by means of giving a general  outline of the key ideas behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We cannot hope to do much better than this since there  are a variety of techniques for obtaining loop equations, as evidenced by the term ‘loop  equations’ being somewhat interchangeable with ‘Ward identities’, ‘Virasoro constraints’,  ‘Tutte’s equations’ and ‘Schwinger–Dyson equations’ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [243] and Chapters 8, 10, 16,  17, 26, and 30 of the handbook [14]), as well as the method of constructing and recursively  solving loop equations being related to the topological recursion [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This relation to the  topological recursion is briefly reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, while §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 contains a survey of how  loop equations for the Gaussian, Laguerre, and Jacobi β ensembles (defined in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) were  derived in [108], [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following the introductory content of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, we demonstrate  the loop equation formalism in more detail, using an alternative approach to that discussed  in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, by applying it to the (N, N) antisymmetrised and Hermitised Laguerre ensembles  in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  A Brief Introduction to Loop Equations  As with many tools in mathematics, the loop equation formalism is essentially a sophisticated  application of integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Following the notation of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, let X be an N × N  random matrix drawn from the ensemble E = (S, P), let p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) be the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' of  the eigenvalues {λi}N  i=1 of X, let ⟨ · ⟩ denote an average with respect to this eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', and let ⟨ · ⟩P(X) denote an average with respect to the matrix p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The main  idea behind the loop equation formalism [243], [19] is to integrate the total derivative of  207  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  the product of P(X) and a well-chosen product of functions, F(X) = ∏i Fi(X) say, in two  different ways: the first is to observe that the integral vanishes by the fundamental theorem  of calculus due to the function F(X) having been chosen such that its product with P(X) is  zero on the boundary ∂S of the domain of integration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the second is to expand the derivative  under the integral sign using the Leibniz product rule and then integrate each of the terms  separately so as to obtain the identity  0 =  �  S  � d  dX P(X)  �  F(X) dX + ∑  i  �� d  dX Fi(X)  � F(X)  Fi(X)  �  P(X)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  For this identity to be relevant to our goal, one of the integrals on the right-hand side must  be equal to either the mixed moment (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn :=  �  N  ∏  i=1  Tr Xki  �  P(X)  =  �  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  λk1  i1 λk2  i2 · · · λkn  in  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  with n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N generic, or the unconnected n-point correlator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) :=  �  N  ∏  i=1  Tr  1  xi − X  �  P(X)  =  �  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x1 − λi1) · · · (xn − λin)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  with generic n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It turns out that good choices of F(X) are those such that the average  �  d  dX F(X)  �  P(X)  (recall from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that the average of an operator with respect to a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) is the integral  of that operator applied to said p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') can be seen to be slight perturbations of either the  average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If, after choosing such an F(X), all of the integrals in the identity  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) can be written in terms of mixed moments mk′  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',k′  n′ with at least one such integral  being mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn exactly, alternatively unconnected correlators Un′(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn′) with at least  one integral being Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn), we say that this identity is a loop equation on the mixed  moments, respectively unconnected correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In practice, at least one of the integrals in  the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) cannot be expressed in the necessary way, so one must repeat this exercise  to find simpler complementary identities expressing these undesirable integrals in terms of  mixed moments or unconnected correlators, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Assuming we succeed in finding the identities described above, we would then have a  set of loop equations on the unconnected correlators (one for each n ∈ N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' note that if one  instead has a set of loop equations on the mixed moments, multiplying the nth loop equation  208  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  by x−k1−1  1  · · x−kn−1  n  and summing over k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 results in the associated loop equation  on the unconnected correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Such a set of loop equations on the unconnected correlators  can be transformed into a corresponding set of loop equations on the connected correlators  through the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31), which we repeat here for convenience,  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) = Un(x1, Jn) − ∑  ∅̸=J⊆Jn  Wn−#J(x1, Jn \\ J)U#J(J),  Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  If we also have a large N expansion (henceforth also referred to as a topological or genus  expansion) of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) available, we may substitute this expansion into the loop  equations for the Wn and then equate terms of like order in N to extract loop equations on  the Wl  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As we will see in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, it is this final set of loop equations that  are exactly solvable through a suitable recursive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Up until now, we have formulated our discussion in terms of averages with respect to  the matrix p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we can see from equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) above that  we have the option to work with either the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) or the matrix  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X), which is equivalent to working with univariate p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s on matrix entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (Note  that in the latter case, we do not necessarily mean p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s on the entries of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' if X is a product  of simpler matrices, it is often beneficial to work with the entries of the factors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') Both  have their advantages and disadvantages: In the first convention, one has the benefit of  only needing to work with the N variables λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN, but may have to deal with unwieldy  eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s — this is certainly the case for our matrix product ensembles (at least  when taking m > 1, as we hope to eventually do);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand,  the second convention requires us to work with many more variables, but each of these  variables may be drawn from relatively simple distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since the Hermitised and antisymmetrised Laguerre ensembles are specified in terms of  Ginibre matrices, whose entries are normally distributed, we opt to study these ensembles  using averages with respect to the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To supplement our development,  we review in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 how averages with respect to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) have been  used to study the Gaussian, Laguerre, and Jacobi β ensembles — recall that these ensembles  do not have amenable matrix p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='s outside of the β = 1, 2, 4 regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We continue this  review in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, where we discuss how the topological recursion [110] results from using  residue calculus to simplify the GUE loop equations to their most aesthetic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  209  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Loop equations for the classical β ensembles  Loop equations for the general β ensembles, which correspond to the general-β eigenvalue  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) with w(λ) =: e−κNV(λ) no longer constrained to be classical (recall that  κ = β/2), were first obtained in the 2006 work [62] of Chekhov and Eynard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Writing the  potential V(λ) as the formal power series  V(λ) = 1 +  ∞  ∑  k=1  tkλk,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  the loop equations of [62] were formulated in terms of the linear integral operator ˆK[ · ] and  functional derivative  ∂  ∂V(x) defined by (see also [19])  ˆK [ f (x)] =  �  γ  dξ  2πi  V′(ξ)  x − ξ f (ξ),  ∂  ∂V(x) = −  ∞  ∑  k=1  1  xk+1  ∂  ∂tk  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  where x lies outside the integration contour γ, which circles the branch cuts of the resolvent  W1(ξ) in a positive direction, and the so-called loop insertion operator  ∂  ∂V(x) is such that the  n-point connected correlator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) can be (formally) written as  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  ∂  ∂V(xn)  ∂  ∂V(xn−1) · · ·  ∂  ∂V(x2)W1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  These operators posed subtle challenges when it came to solving the relevant loop equations  in practice, so about three years later, Eynard and Marchal [108] derived a more tractable  reformulation of said loop equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' There, it was shown through a consideration of the  infinitesimal change of variables  λi �→ λi + ϵ  1  x1 − λi  + O(ϵ2)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  in the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) that the relevant connected correlators (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) satisfy, for each n ⩾ 1,  the loop equation (see also [55], [43], [319])  κ ∑  J⊆Jn  W#J+1(x1, J)Wn−#J(x1, Jn \\ J) + κWn+1(x1, x1, Jn) + (κ − 1) ∂  ∂x1  Wn(x1, Jn)  = κN  �  V′(x1)Wn(x1, Jn) − Pn(x1, Jn)  � −  n  ∑  i=2  ∂  ∂xi  �Wn−1(x1, Jn \\ {xi}) − Wn−1(Jn)  x1 − xi  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  where Jn is as in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), #S denotes the size of S, and  Pn(x1, Jn) :=  �  N  ∑  i1=1  V′(x1) − V′(λi1)  x1 − λi1  N  ∑  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x2 − λi2) · · · (xn − λin)  �  conn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  210  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  Here, the connected average ⟨ · ⟩conn relates to the usual average ⟨ · ⟩ in the same way that the  connected and unconnected correlators relate to each other via equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, it is straightforward to specialise the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) to the case of the Gaussian  β ensemble: one need only observe that taking V(λ) to be the quadratic potential λ2 in  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) shows that (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [319, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1])  P(G)  n  (x1, Jn) =  �  2N  N  ∑  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x2 − λi2) · · · (xn − λin)  �  conn  = 2Nχn=1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  where we recall that the indicator function χA equals one when A is true and is otherwise  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, if we let ˜W(G)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) denote the connected n-point correlator of the global  scaled Gaussian β ensemble with eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (apply the scaling λi �→  √  κNλi, in  keeping with Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6, to the j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81) with w(λ) = e−λ2)  ˜p(G)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' β) = (κN)  N  2 (κ(N−1)+1)  N (G)  N,β  N  ∏  i=1  e−κNλ2  i |∆N(λ)|β,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  substituting equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) into the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) shows that  κ ∑  J⊆Jn  ˜W(G)  #J+1(x1, J) ˜W(G)  n−#J(x1, Jn \\ J) + κ ˜W(G)  n+1(x1, x1, Jn) + (κ − 1) ∂  ∂x1  ˜W(G)  n  (x1, Jn)  = 2κN  �  x1 ˜W(G)  n  (x1, Jn) − Nχn=1  �  −  n  ∑  i=2  ∂  ∂xi  � ˜W(G)  n−1(x1, Jn \\ {xi}) − ˜W(G)  n−1(Jn)  x1 − xi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  Moreover, invoking Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to further substitute the expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  ˜W(G)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) = N2−n  ∞  ∑  l=0  W(G),l  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  Nl  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) and equating terms of equal order in N yields, for n ⩾ 1 and l ⩾ 0, the  (n, l) loop equation on the correlator expansion coefficients,  κ ∑  J⊆Jn  l  ∑  k=0  W(G),k  #J+1 (x1, J)W(G),l−k  n−#J  (x1, Jn \\ J) + κW(G),l−2  n+1  (x1, x1, Jn)  = (1 − κ) ∂  ∂x1  W(G),l−1  n  (x1, Jn) + 2κx1W(G),l  n  (x1, Jn) − 2κχn=1,l=0  −  n  ∑  i=2  ∂  ∂xi  �  W(G),l  n−1 (x1, Jn \\ {xi}) − W(G),l  n−1 (Jn)  x1 − xi  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  with W(G),−2  n  , W(G),−1  n  := 0 for all n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It can readily be seen that for any given m ∈ N, the  set of loop equations obtained by setting n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , m in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) involves more than  211  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  m correlators and is thus unsolvable, but the same is not true for equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed,  setting (n, l) = (1, 0) in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) produces the so-called spectral curve [109]  �  W(G),0  1  (x1)  �2  − 2x1W(G),0  1  (x1) + 2 = 0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  which can be easily solved to recover the Stieltjes transform W(G),0  1  (x1) = x1 −  �  x2  1 − 2 of  the Wigner semi-circle law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Having this expression at hand then enables one to  solve equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) with (n, l) = (1, 1) to obtain W(G),1  1  (x1), which then allows for the  computation of W(G),0  2  (x1, x2) through the (2, 0) loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, as  mentioned in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, the (n, l) loop equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) can be solved in a triangular recursive  manner and this is moreover true for all analogous Wl  n loop equations discussed in this  chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The entries of the table below specify the order in which said loop equations must  be solved to eventually compute W10  1  — given enough computation power and time, Wl  n can,  in theory, be computed for any n ⩾ 1 and l ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  l  0  1  2  3  4  5  6  7  8  9  10  n  Wl  n  1  1  2  4  6  9  12  16  20  25  30  36  2  3  5  8  11  15  19  24  29  35  3  7  10  14  18  23  28  34  4  13  17  22  27  33  5  21  26  32  6  31  It turns out that obtaining Wl  n loop equations of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) for the Laguerre and  Jacobi β ensembles is more involved than in the Gaussian case, especially if one wishes to  allow the Laguerre and Jacobi parameters a, b to vary with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is because the potentials  V(λ) corresponding to the global scaled Laguerre and Jacobi β ensembles (compare the  weight w(λ) = e−κNV(λ) to the results of applying the scalings of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6 to the weights  given in equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)) are respectively  V(L)(λ) = λ − a  κN log(λ),  V(J)(λ) = − a  κN log(λ) − b  κN log(1 − λ),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  which means that the auxiliary function Pn(x1, Jn) specified by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) is noticeably  more complicated than seen in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, loop equations for the Laguerre  212  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  and Jacobi β ensembles were first made explicit in the 2017 work [142] of Forrester, the  present author, and Witte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The proofs therein used an adaptation of Aomoto’s method [22]  for proving the Selberg integral formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), which differs slightly in spirit from the  derivation of [108] reviewed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (Note that the same adaptation of Aomoto’s method was  also used by Witte and Forrester in 2015 to obtain loop equations for the circular β ensembles  [320].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') Let us now review how Aomoto’s method was used in [142] to tackle the problem of  specifying Pn(x1, Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In keeping with the general formalism prescribed at the beginning of this section, we  begin by considering the average  �  N  ∑  i1=1  ∂  ∂λi1  1  x1 − λi1  N  ∑  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x2 − λi2) · · · (xn − λin)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  with respect to the global scaled eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) of either the Gaussian,  Laguerre, or Jacobi β ensemble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' recall that the differential operator  ∂  ∂λi1 acts on the product  of all terms on its right in the integral formulation of this average, including ˜p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The equivalent of making the change of variables (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) in the definition of the unconnected  correlators is to apply integration by parts to the above average in order to obtain an identity  of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Temporarily taking a, b > 0 so that the weights w(L)(λ), w(J)(λ) equal  (or limit to) zero at the boundary of the domain of integration (and later relaxing this  requirement to a, b > −1 using analytic continuation), we see that this average must vanish  by the fundamental theorem of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the other hand, using the Leibniz product rule  to expand the integrand shows that  0 =  �  N  ∑  i1=1  ∂  ∂λi1  1  x1 − λi1  N  ∑  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x2 − λi2) · · · (xn − λin)  �  = κ ˜Un+1(x1, x1, Jn) + (κ − 1) ∂  ∂x1  ˜Un(x1, Jn) −  n  ∑  i=2  ∂  ∂xi  � ˜Un−1(x1, Jn \\ {xi}) − ˜Un−1(Jn)  xi − x1  �  − κN  �  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  V′(λi1)  (x1 − λi1) · · · (xn − λin)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  where ˜Un denotes the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) with respect to the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˜p(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) of  the global scaled classical β ensemble at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Gaussian case, V′(λi1) = 2λi1 =  2x1 − 2(x1 − λi1), so the average on the bottom line of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) simplifies to  2x1 ˜U(G)  n  (x1, Jn) − 2N ˜U(G)  n−1(Jn) and thus equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) can be seen to be a loop equa-  213  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  tion on the ˜U(G)  n  which, upon the use of an inductive argument based on equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  (see [142, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A]), is equivalent to the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) on the ˜W(G)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the Laguerre  case, we have from equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) that V′(λi1) = 1 − a/(κNλi1), so the aforementioned  average on the bottom line of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) instead simplifies to  �  1 −  a  κNx1  �  ˜U(L)  n (x1, Jn) −  a  κNx1  �  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  λi1(x2 − λi2) · · · (xn − λin)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  which is not immediately expressible in terms of the ˜U(L)  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we search for a companion  to the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) whose expansion via Aomoto’s method produces an identity of the  form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) that is simpler than equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18), but contains the troublesome average seen  in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It was shown in [142] that it is sufficient to consider the average  �  N  ∑  i1=1  ∂  ∂λi1  N  ∑  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x2 − λi2) · · · (xn − λin)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  which, by a replication of the above arguments, is equal to both zero and  a  �  N  ∑  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  λi1(x2 − λi2) · · · (xn − λin)  �  − N ˜U(L)  n−1(Jn) −  n  ∑  i=2  ∂  ∂xi  ˜U(L)  n−1(Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  Thus, combining equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) with the fact that the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) equals zero  results in a loop equation on the ˜U(L)  n  that further transforms via equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) into a loop  equation on the ˜W(L)  n  [142, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This loop equation is the same as equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8),  except with each Wn′ rewritten as ˜W(L)  n′  and with the replacement  κN  �  V′(x1)Wn(x1, Jn) − Pn(x1, Jn)  � �→  �  κN − a  x1  �  ˜W(L)  n (x1, Jn)  − χn=1  κN2  x1  − 1  x1  n  ∑  i=2  ∂  ∂xi  ˜W(L)  n−1(Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  Note that the second line of the right-hand side of the above is exactly Pn(x1, Jn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) in  the Laguerre case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, to obtain loop equations for the Jacobi β ensemble, one repeats the above exercise  of applying integration by parts to the averages (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20), now with respect to  the eigenvalue j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p(J)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, it is once again observed [142] that  this does not result in an equation that can be written fully in terms of the U(J)  n  (we do not  214  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  write ˜U(J)  n  nor ˜W(J)  n  since no scaling is required in the Jacobi case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see the discussion below  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) — one must also consider the third average  �  N  ∑  i1=1  ∂  ∂λi1  λi1  N  ∑  i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',in=1  1  (x2 − λi2) · · · (xn − λin)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  Combining the identity resulting from applying integration by parts to this average with the  Jacobi analogues of the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) and that obtained by setting the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  to zero then gives a loop equation on the U(J)  n , which is again equivalent to a loop equation  on the W(J)  n  due to the aforementioned inductive argument involving the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Like in the Gaussian and Laguerre cases, this latter loop equation has the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) with  each Wn′ rewritten as W(J)  n′  and with the replacement [142, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4]  κN  �  V′(x1)Wn(x1, Jn) − Pn(x1, Jn)  � �→  �  b  1 − x1  − a  x1  �  W(J)  n (x1, Jn) +  n − 1  x1(1 − x1)W(J)  n−1(Jn)  −  χn=1  x1(1 − x1)[(a + b + 1)N + κN(N − 1)]  +  1  x1(1 − x1)  n  ∑  i=2  xi  ∂  ∂xi  W(J)  n−1(Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  As we move on, let us mention that in contrast to the Gaussian case, the Wl  n loop equations  for the Laguerre and Jacobi β ensembles depend on how a, b vary with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless, the  spectral curves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the analogues of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)) are quadratic in W0  1(x1) [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  From loop equations to the topological recursion  The topological recursion was first given in its present form in the 2009 work [110] of  Eynard and Orantin, where it arose as a refinement of the loop equations derived about five  years earlier [105], [109] for the 1-Hermitian matrix model with polynomial potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  so-called 1-Hermitian matrix model is the ensemble of eigenvalues with j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  p(1HMM)(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , λN) =  1  N (1HMM)  N  N  ∏  i=1  e−NV(λi)|∆N(λ)|2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  where the potential V(λ) is as in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) [19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' note that this ensemble is the same as  the general β ensembles discussed in the previous subsection, but with β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us now  sketch the derivation of the topological recursion for the 1-Hermitian matrix model (with  V(λ) constrained to be polynomial) given in the work [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  215  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Setting κ = β/2 = 1 in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) produces the loop equation on the connected  n-point correlators of the 1-Hermitian matrix model,  ∑  J⊆Jn  W#J+1(x1, J)Wn−#J(x1, Jn \\ J) + Wn+1(x1, x1, Jn)  = N  �  V′(x1)Wn(x1, Jn) − Pn(x1, Jn)  �  −  n  ∑  i=2  ∂  ∂xi  �Wn−1(x1, Jn \\ {xi}) − Wn−1(Jn)  x1 − xi  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  where Jn is as in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and Pn(x1, Jn) is as specified by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us  now restrict the potential V(λ) to be independent of N so that substituting the topological  expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) into the above loop equation shows that Pn(x1, Jn) has an analogous  expansion of the form  Pn(x1, Jn) = N2−n  ∞  ∑  l=0  Pl  n(x1, Jn)  Nl  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  Thus, substituting both expansions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) and equating terms  of like order in N then produces the loop equation on the correlator expansion coefficients,  ∑  J⊆Jn  l  ∑  k=0  Wk  #J+1(x1, J)Wl−k  n−#J(x1, Jn \\ J) + Wl−2  n+1(x1, x1, Jn)  = V′(x1)Wl  n(x1, Jn) − Pl  n(x1, Jn) −  n  ∑  i=2  ∂  ∂xi  �  Wl  n−1(x1, Jn \\ {xi}) − Wl  n−1(Jn)  x1 − xi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  In particular, taking (n, l) = (1, 0) yields the spectral curve (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)):  �  W0  1(x1)  �2 − V′(x1)W0  1(x1) + P0  1 (x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  Having assumed V(λ) to be polynomial, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) tells us that P1(x1) must also be  polynomial in x1 of degree one less than V′(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) can be shown (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [111]) to have solution of the form  W0  1(x1) = 1  2  �  V′(x1) ± M(x1)  �  σ(x1)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  where M(x1) and σ(x1) are polynomials such that the latter is of even degree, say 2s, with  all roots distinct and M(x1)2σ(x1) = (V′(x1))2 − 4P0  1 (x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From the perspective of random  matrix theory, one must take the negative sign in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) so that W0  1(x1) ∼ 1/x1 as  x1 → ∞, in line with, say, equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  216  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  The first step in simplifying the above loop equations is to circumvent the multivalued  nature of W0  1(x1) by viewing it as a single-valued holomorphic function on an appropriate  two-sheeted cover of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This cover can be constructed by taking two copies  of the complex plane, gluing them along the s branch cuts of W0  1(x1) (which are the s  intervals along which the degree-2s polynomial σ(x1) is negative) and then adding a point  at infinity to each of the two complex planes so that the resulting surface is a genus s − 1  compact Riemann surface Σ [111];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Such a covering space Σ comes  equipped with a holomorphic projection x : Σ → C ∪ {∞} such that for all x1 ∈ C ∪ {∞}  not a root of σ(x1), there exist two distinct points z, z′ ∈ Σ such that x(z) = x(z′) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  More importantly, there exists a single-valued meromorphic function y : Σ → C such that  whenever z, z′ ∈ Σ are such that z ̸= z′, but x(z) = x(z′), y(z) and y(z′) are the two values  of W0  1(x(z)) given by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Here on out, let us work in the one-cut case, which  corresponds to assuming that σ(x1) is quadratic with two distinct roots, u < v say, and thus  Σ is the Riemann sphere CP1 (being genus zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1: On the left, we have two copies of the complex plane with their branch cuts [u, v]  identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' note that the blue plane is obtained from the beige plane through a reflection  in the real axis to ensure that if one approaches the branch cut from the beige positive  (negative) half plane, one leaves the branch cut by entering the blue negative (positive) half  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the middle image, we have opened up the branch cuts into circles and pulled  them together so as to form a tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Compactifying by adding (orange) points at infinity to  the beige and blue planes results in the Riemann sphere on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that if there  were more cuts, there would be more tubes which would end up becoming handles in the  final surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For each x1 ∈ C ∪ {∞} \\ {u, v}, there exists a z in the beige hemisphere and  z′ in the blue hemisphere such that x(z) = x(z′) = x1 and y(z), y(z′) are the two values of  W0  1 (x1) given in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content="  217  Im z  Im z  7  7  u  Re'z  Re'z  u  V  u  V  Rez'  Rez'  Im z'  Im z'CHAPTER 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  In the one-cut case, one can use the Joukowsky transform to see that when the endpoints  of our branch cut are u < v,  x(z) = v − u  4  �  z + 1  z  �  + u + v  2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  has the properties required of the projection map described above: this mapping has exactly  u, v as branch points and every x1 ∈ C ∪ {∞} \\ {u, v} has two preimages under x (note that  x(−1) = u and x(1) = v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Furthermore, we can choose a sign in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) to write  y(z) = 1  2  �  V′(x(z)) − v − u  4  M(x(z))  �  z − 1  z  ��  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  observe that for all z ∈ Σ = CP1, x(z) = x(1/z), so if z ̸= ±1 is a preimage of x1, y(z)  and y(1/z) are the two values of W0  1(x1) given in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This reformulation  of (x1, W0  1(x1)) as the coordinates (x(z), y(z)) on Σ extends naturally to the spectral curve  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) and, indeed, to the general (n, l) loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, since the (n, l) loop  equation specifying Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) is linear in this correlator expansion coefficient (when  (n, l) ̸= (1, 0)), we see by induction that each of these correlators have the same branching  structure as W0  1(x1) and so can be viewed as single-valued holomorphic functions on Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, we are led to rewrite equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) in terms of the coordinates (x(z), y(z)) on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In fact, to facilitate upcoming residue calculus, we are encouraged to introduce differential  forms on Σ, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, let us define for z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , zn ∈ Σ and (n, l) ̸= (1, 0), (2, 0),  ω0  1(z1) := y(z1)dx(z1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  ω0  2(z1, z2) :=  �  W0  2(x(z1), x(z2)) +  1  (x(z1) − x(z2))2  �  dx(z1)dx(z2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  ωl  n(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , zn) := Wl  n(x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn))dx(z1) · · · dx(zn),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  where equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) is a specialisation of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) based on the replacement of  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) is chosen to differ from equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) in a way that ensures that ω0  2(z1, z2) coincides with the Bergman kernel (also known  as the fundamental differential of the second kind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [110], [111]) dz1dz2/(z1 − z2)2  on CP11, and the Wl  n(x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn)) within equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) is to be interpreted as the  1It was first shown in [18], [19] that for the one-cut 1-Hermitian matrix model, W0  2 has the universal form  implied by the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) being equal to the Bergman kernel on CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This was also  observed [319], [142] for the classical β ensembles discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  218  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  single-valued function on (CP1)n obtained from the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) after it has been  rewritten in terms of the coordinates (x(z), y(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  With definitions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) in hand and noting that V′(x1) − 2W0  1(x1) should be  replaced by y(1/z1) − y(z1) to be consistent with our choice of replacing W0  1(x1) with y(z1),  we rewrite equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) for (n, l) ̸= (1, 0), (2, 0) as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [107, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3])  ωl  n(z1, J′  n) =  1  ω0  1(1/z1) − ω0  1(z1)  �  ωl−2  n+1(z1, z1, J′  n) + ∑  J⊆J′  n  0⩽k⩽l  ωk  #J+1(z1, J)ωl−k  n−#J(z1, J′  n \\ J)  �  −  1  ω0  1(1/z1) − ω0  1(z1)  �  χn=1,l=2 lim  ξ→z1  dx(z1)dx(ξ)  (x(z1) − x(ξ))2  + 2  n  ∑  i=2  dx(z1)dx(zi)ωl  n−1(z1, J′  n \\ {zi})  (x(z1) − x(zi))2  �  + dx(z1) · · · dx(zn)  y(1/z1) − y(z1)  �  Pl  n(x(z1), Jn) +  n  ∑  i=2  4  v − u  z2  i  z2  i − 1  × ∂  ∂zi  �  Wl  n−1(x(z1), Jn \\ {x(zi)}) − Wl  n−1(Jn)  x(z1) − x(zi)  � �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  where we now have Jn = (x(z2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn)), J′  n = (z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , zn), and the sum ∑◦ excludes the  terms corresponding to (k, J) = (0, {zi}), (l, J′  n \\ {zi});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the terms of the second and third line  arise from the fact that occurrences of ω0  2 in the top line must be accompanied by the extra  term shown in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34), as compared to equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now, replace z1 by ξ in  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36), multiply both sides by a half times the meromorphic differential [110]  dSξ,1/ξ(z1) :=  � z=ξ  z=1/ξ ω0  2(z1, z) =  �  1  z1 − ξ −  1  z1 − 1/ξ  �  dz1,  ξ ∈ CP1 \\ {±1},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  and take the sum of the residues at ξ = ±1 on both sides of the resulting equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  shows that the left-hand side simply reduces as  1  2 ∑  a=±1  Res  ξ=a dSξ,1/ξ(z1)ωl  n(ξ, J′  n) = 1  2 ∑  a=±1  Res  ξ=a  �  1  z1 − ξ −  1  z1 − 1/ξ  �  ωl  n(ξ, J′  n) dz1  = ∑  a=±1  Res  ξ=a  1  z1 − ξ ωl  n(ξ, J′  n) dz1 = Res  ξ=z1  1  ξ − z1  ωl  n(ξ, J′  n) dz1 = ωl  n(z1, J′  n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  the second equality is obtained by changing variables ξ �→ 1/ξ in Res  ξ=aωl  n(ξ, J′  n)/(z1 − 1/ξ)  and using the fact that ωl  n(1/ξ, J′  n) = −ωl  n(ξ, J′  n) (this can be checked inductively through  219  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) [110], [107, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3]), while the third equality follows from the fact that  the sum of all residues of a meromorphic form on CP1 must vanish (again, an inductive  argument via equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) shows that ωl  n(ξ, J′  n) only has residues at ξ = ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On the  other hand, the right-hand side simplifies remarkably to  ∑  a=±1  Res  ξ=a K(ξ, z1)  �  ωl−2  n+1(ξ, 1/ξ, J′  n) + ∑  J⊆J′  n  0⩽k⩽l  ωk  #J+1(ξ, J)ωl−k  n−#J(1/ξ, J′  n \\ J)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  where we define the recursion kernel as  K(ξ, z1) :=  dSξ,1/ξ(z1)  2(ω0  1(ξ) − ω0  1(1/ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  The simplified form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) results from the fact that all of the terms below the first line of  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) either do not have residue at z1 = ±1 or the sum of their non-zero residues  vanish by symmetry arguments [107, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, setting the left- and right-hand sides  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) equal to each other gives us the (Eynard–Orantin) topological recursion  for the one-cut 1-Hermitian matrix model with polynomial potential:  ωl  n(z1, J′  n) =  ∑  a=±1  Res  ξ=a K(ξ, z1)  �  ωl−2  n+1(ξ, 1/ξ, J���  n) + ∑  J⊆J′  n  0⩽k⩽l  ωk  #J+1(ξ, J)ωl−k  n−#J(1/ξ, J′  n \\ J)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  Observe that this topological recursion formula can be seen to be a simplification of the loop  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) obtained by essentially replacing the right-hand side by Wl  n(x1, Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note  that this means that knowledge of the Pl  n(x1, Jn) is not needed to compute the ωl  n — recall  from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that without the topological recursion formula available, other techniques, such  as Aomoto’s method, must be used to deal with Pl  n(x1, Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us now explain why the topological recursion is so named.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Assume for simplicity that  the potential V(λ) is the Gaussian potential λ2 (it has however been known since the 1970s  [179], [54], [36] that the following argument applies formally to more general potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [107, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, by the discussion of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, the Wl  n are zero for odd values of l and  are otherwise generating functions for the compact, connected, orientable ribbon graphs of  genus l/2 that can be built from n polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we should take l to be even in equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) and discard terms in the sum involving odd values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, in keeping with  220  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A Brief Introduction to Loop Equations  the discussion involving Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 and that following Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10, it is convenient to  visualise ωl  n as a genus l/2 surface with n holes or punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We can then interpret the  topological recursion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) in the following diagrammatic fashion [109], [110]:  The process of multiplying by the recursion kernel K(ξ, z1) and then taking the sum of the  residues at ξ = ±1 is represented by the gluing of a pair of pants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', a sphere with three  holes) to either two holes of a genus l/2 − 1 surface with n + 1 holes to increase the genus  and lower the number of holes by one each, or to one hole each of two separate surfaces in  order to form a genus l/2 surface with n holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The recursion thus  runs over the Euler characteristic χ = 2 − l + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2: On the left, we have two homeomorphic genus 2 surfaces with 1 hole represent-  ing the term ω4  1(z1) in the left-hand side of the topological recursion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) with  (n, l) = (1, 4) (having taken n to be the number of holes and l/2 to be the genus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the top  row, we see that gluing a pair of pants (in pink) representing the operator ∑a=±1 Res  ξ=aK(ξ, z1)  to a genus 1 surface with 2 holes representing ω2  2(ξ, 1/ξ) produces the surface representing  ω4  1(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the bottom row, we see that gluing such a pair of pants to two genus 1 surfaces  with 1 hole each produces the surface on the bottom left — the two blue surfaces on the  bottom right represent ω2  1(ξ) and ω2  1(1/ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The diagrammatic interpretation of the topological recursion described above can be  traced back to Tutte’s equation and recursion [302], [309] (see also [107, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3] for a textbook  treatment) on the coefficients of the genus expansions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21) of the GUE mixed cumulants  c(GUE)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the language of ribbon graphs (rather than topological maps, as considered  by Tutte), the idea behind Tutte’s recursion is to simply observe that a given connected,  orientable ribbon graph with n polygons can be obtained from a similar such ribbon graph  with n + 1 polygons by shrinking a particular ribbon to identify its ends so that the two  221  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  relevant polygons merge into one — if instead deleting said ribbon results in a connected  (disconnected) ribbon graph with two unpaired polygon edges, then we have a ribbon graph  analogue of the decomposition displayed in the top (bottom) row of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The key insight of the seminal work [110] was that the mechanism underlying the  topological recursion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) is common to many enumerative problems in random  matrix theory and algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In the algebro-geometric setting, one abstracts by  taking the spectral curve to be a triple (Σ, x, y) of a Riemann surface Σ and two meromorphic  functions x, y : Σ → C ∪ {∞}, defines ω0  1(z1) = y(z1)dx(z1), defines ω0  2(z1, z2) to be the  Bergman kernel on Σ2, and then recursively defines all other ωl  n through equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  reformulated in terms of the genus g = l/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', setting ωl′  n = 0 whenever l′ is odd) and with  the sum over a = ±1 replaced by a sum over all branch points, equivalently zeroes of the  differential form dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, for particular choices of initial data (Σ, x, y, ω0  2), the topological  recursion has been shown to govern, for example, Gromov–Witten invariants of CP1, various  Hurwitz numbers, dessins d’enfants, the intersection numbers mentioned at the beginning  of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, and Weil–Petersson volumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [110], [91], [265], [61] and references  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, Mirzakhani’s groundbreaking recursion [245], [246] for Weil–Petersson  volumes has been shown to be an instance of the topological recursion [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since the work of Eynard and Orantin [110], the topological recursion has been generalised  and reformulated in a variety of ways: Eynard and Marchal [108] showed that the topological  recursion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) can be applied to general β ensembles upon redefining the  recursion kernel K(ξ, z1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) in a suitable β-dependent way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Bouchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [45], [46]  studied generalisations of the topological recursion involving higher order spectral curves  by introducing terms on the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Do and Norbury [86]  and Chekhov and Norbury [63] have studied aspects of the topological recursion when the  spectral curve is irregular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', the eigenvalue density has hard edges) — in particular, they  showed that the connected correlators of the LUE with a = 0 and the Legendre unitary  ensemble, equivalently the sJUE with a = b = 0, obey the topological recursion (like [142],  their results provide an alternative to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 for computing the  moment expansion coefficients of the LUE with a = 0 and the Legendre unitary ensemble);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Kontsevich and Soibelman [212] have recently reformulated the topological recursion by  replacing the initial data with a certain set of tensors derived from ω0  1 and ω0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  222  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Loop Equations for the Antisymmetrised Laguerre Ensemble  In this section, we derive loop equations for the mixed moments ˜m(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), the uncon-  nected n-point correlators ˜U(J1)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), the corresponding connected correlators  ˜W(J1)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), and finally some correlator expansion coefficients W(J1),l  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) of the global scaled (N, N) antisymmetrised Laguerre ensemble, whose definition we  recall from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To improve clarity, we begin by introducing some new notation  (local to this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, take N to be an even integer and introduce the scaled real N × N Ginibre matrix  X ∈ MN×N(R) with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  P(X) =  � N  2π  �N2/2  exp  �  − N  2 Tr(XTX)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  Moreover, denote the elementary antisymmetric matrix JN (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) by simply the letter J and  define B := XTJX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, X is statistically equal to the scaled real Ginibre matrix ˜G = ˜G1 of  Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, and therefore relates to the real Ginibre matrix G of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  through the equation X = √  2/NG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, the antisymmetric matrix product B = XTJX  equals the matrix ˜J1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='60) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and is thus related to the unscaled matrix  J1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) according to the equality B = 2J1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Next, let ⟨ · ⟩ denote averages with respect to the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) and recall that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  ⟨∂Xab f (X)⟩ should be read as “take the partial derivative of the product f (X)P(X) with  respect to Xab and integrate the result over MN×N(R)”, while ⟨{∂Xab f (X)}⟩ reads “multiply  the partial derivative of f (X) with respect to Xab by P(X) and then integrate the result over  MN×N(R)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N, the mixed moments and unconnected n-point correlators of  B are respectively specified by (recall equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17))  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  n  ∏  i=1  Tr Bki  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  ∞  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn=0  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  xk1+1  1  · · xkn+1  n  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  note that whenever any of the ki are odd, the trace of the antisymmetric matrix Bki is zero,  so mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The connected n-point correlator Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) is most easily specified  in terms of the above through equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), but has an alternative charcterisation as a  223  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  generating function of the mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn — which are defined implicitly via the  moment-cumulants relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) — through the expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  ∞  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn=0  ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  xk1+1  1  · · xkn+1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  These quantities are the same as the scaled analogues discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we have only  simplified notation by omitting the tilde and the superscript (J1) so that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', ˜c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn is now  written simply as ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' If we use the superscript (J1), without a tilde, to denote statistics  of the unscaled ensemble represented by the matrix J1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), we have  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  � 2  N  �k1+···+kn  m(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  � 2  N  �k1+···+kn  c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  � N  2  �n  U(J1)  n  (Nx1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nxn/2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  � N  2  �n  W(J1)  n  (Nx1/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , Nxn/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  Now, inserting the genus expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65) for the mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = ˜c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) shows that Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) admits a large N expansion of the form  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) and we may thus implicitly define the correlator expansion coefficients Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, let us adopt the Einstein summation convention, meaning that repeated matrix  indices in a product of matrix entries are to be summed over 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example, AabBbc  denotes the sum over b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N of the product of the (a, b) entry of A and (b, c) entry of  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, AabBbc = (AB)ac, the (a, c) entry of the matrix product AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In particular, we must  take care to remember that Jaa = Tr(J) = 0 and ∂XabXab = ∑N  a,b=1 1 = N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As discussed at the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, the first step in deriving our loop equations  is to use integration by parts on a suitable perturbation of the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) (recall that a  perturbation of the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) can also serve as an alternative starting point) in order  to produce an identity of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us recall from the discussion of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  (particularly that preceding Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) that traces Tr Bk can be conveniently represented  by k-gons formed by chaining together edges labelled by entries of B into a closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The  intuition behind the first average we consider in our development is that we would like to  224  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  delete a term in the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) specifying mk1+1,k2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to cut open the loop representing  Tr Bk1+1 and then reintroduce that same term to join the loop back up, except that the  reintroduced term is now extracted from the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) by way of acting on said  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' with an appropriate differential operator2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us write Tr Bk1+1 = (Bk1)adBda (using the Einstein summation convention) and  then erase the last term so that we are left with (Bk1)ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, let us define In := (k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn)  and use the shorthand Tr BIn := Tr Bk2 Tr Bk3 · · · Tr Bkn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have that  ∂(XT)ab(JT)bc∂Xcd exp  �  − N  2 Tr(XTX)  �  = N2Bda exp  �  − N  2 Tr(XTX)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  and, consequently,  �  (Bk1)ad Tr BIn∂(XT)ab(JT)bc∂Xcd  �  = N2mk1+1,In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We give a short proof as a warm-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First note by the Leibniz product rule that  ∂(XT)ab∂Xcd exp  �  − N  2 Tr(XTX)  �  = −N∂(XT)abXcd exp  �  − N  2 Tr(XTX)  �  =  �  N2(XT)abXcd − Nχa=d,b=c  �  exp  �  − N  2 Tr(XTX)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Multiplying this result by (JT)bc gives  �  N2(XTJTX)ad − NJbbχa=d,b=c  �  exp  �  − N  2 Tr(XTX)  �  ,  which reduces to the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) upon recalling that Jbb = Tr J = 0  and observing that (XTJTX)T = XTJX, so (XTJTX)ad = Bda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Pre-multiplying both sides  of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) by (N/2π)N2/2(Bk1)ad Tr BIn and then integrating over MN×N(R) while  keeping track of notation yields equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  From the above, one might suggest that our desired starting point be the average  �  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn  �  , as starting at the average of a full derivative enables us  to use the fundamental theorem of calculus to immediately show that said average is zero —  this is indeed very similar to the initial average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) seen in the loop equation analysis  of the Laguerre and Jacobi β ensembles given in [142], as reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, one  2This intuition is essentially the spirit behind the terminology ‘loop equations’ and is also the reason why  loop equations are sometimes referred to as cut-and-join equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  225  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  soon sees that this path requires us to consider averages of the form  �  Tr(BpXTX) Tr Bq�  ,  which are not immediately expressible in terms of the mixed moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As seen in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1,  this inevitably leads to a game of finding identities that can be combined with the initial  analogue of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) in order to obtain a valid loop equation on the mixed moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Such games can at times prove to be a tedious matter of trial and error, especially for large  products of matrices, and if one does not have much experience in the exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It turns out that, for reasons soon to be made clear, a lot of guesswork can be avoided by  beginning our analysis at the average  ��  ∂(XT)ab + NXba  �  (JT)bc  �  ∂Xcd + N(XT)dc  �  (Bk1)ad Tr BIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  Thus, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 forthcoming, we apply integration by parts to this average to obtain loop  equations on the mixed moments mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn in a relatively straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These loop  equations are then easily transformed into loop equations on the unconnected correlators  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we use equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) to obtain loop equations  for the connected correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) and their expansion coefficients Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  from the Un loop equations of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We then conclude this section with some explicit  computations and discussions in relation to W0  1(x1) and W0  2(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Loop equations on the ˜U(J1)  n  Before unpacking the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11), let us first give the companion to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The partial derivatives of (Bk1)e f with respect to (XT)ab and Xcd (taking e, f to be  arbitrary and possibly equal to each other or one of a, b, c, d) are given by  ∂(XT)ab(Bk1)e f =  ∑  p1+p2=k1−1  �  (Bp1)ea(JXBp2)b f + (Bp1XTJ)eb(Bp2)a f  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  ∂Xcd(Bk1)e f =  ∑  p1+p2=k1−1  �  (Bp1)ed(JXBp2)c f + (Bp1XTJ)ec(Bp2)d f  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  where the sums over p1 + p2 = k1 − 1 are over the range p1 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k1 − 1 with 0 ⩽ p2 ⩽ k1 − 1  set to k1 − p1 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' these sums are empty for k1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For later convenience, let us also mention that  ∂(XT)ab(Bk1XT)e f =  ∑  p1+p2=k1−1  �  (Bp1)ea(JXBp2XT)b f + (Bp1XTJ)eb(Bp2XT)a f  �  + χ f =b(Bk1)ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  226  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First note that by the Leibniz product rule,  ∂(XT)abBgh =  �  ∂(XT)ab(XT)gl  �  (JX)lh + (XTJ)gl  �  ∂(XT)abXlh  �  = χg=a,l=b(JX)lh + (XTJ)glχl=b,h=a = χg=a(JX)bh + χh=a(XTJ)gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, the left-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) decomposes as  ∂(XT)ab(Bk1)e f =  k1−1  ∑  p1=0  (Bp1)eg  �  ∂(XT)abBgh  �  (Bk1−p1−1)h f  =  k1−1  ∑  p1=0  �  (Bp1)ea(JX)bh(Bk1−p1−1)h f + (Bp1)eg(XTJ)gb(Bk1−p1−1)a f  �  ,  which simplifies to the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) upon summing over the repeated  indices g, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To obtain equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) from equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12), simply observe that setting  (c, d) = (b, a) shows that Xcd = (XT)ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14), note that  ∂(XT)ab(Bk1XT)e f =  �  ∂(XT)ab(Bk1)eg  �  (XT)g f + (Bk1)eg  �  ∂(XT)ab(XT)g f  �  =  �  ∂(XT)ab(Bk1)eg  �  (XT)g f + (Bk1)egχg=a,f =b  =  �  ∂(XT)ab(Bk1)eg  �  (XT)g f + (Bk1)eaχ f =b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  substituting equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) with f �→ g into the braces then completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  With these identities in hand, we can now apply integration by parts to the average  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) in two different ways to obtain a precursor to the loop equation on the Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recalling from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that In = (k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn,  the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) simplifies to both the left- and right-hand sides of the following equation:  N2mk1+1,In =  ��  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Expanding the argument of the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) shows that it is given by  ��  ∂(XT)ab + NXba  �  (JT)bc  �  ∂Xcd + N(XT)dc  �  (Bk1)ad Tr BIn  �  =  �  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn  �  + N  �  ∂(XT)ab(JT)bc(XT)dc(Bk1)ad Tr BIn  �  + N  �  Xba(JT)bc∂Xcd(Bk1)ad Tr BIn  �  + N2 �  Xba(JT)bc(XT)dc(Bk1)ad Tr BIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  227  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  The first two terms on the right-hand side of this equation are manifestly zero by the  fundamental theorem of calculus (they are integrals of derivatives of the exponentially  decaying p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' P(X) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) multiplied by polynomials in entries of X, which must converge  to zero at the boundary of the domain of integration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, contracting indices in the  fourth term on the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) while rewriting (JT)bc as Jcb shows  that this term is equal to the left-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, to see that the third  term on the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) vanishes, note that for f (X) a function of  the entries of X, integration by parts shows that  �  Xba(JT)bc∂Xcd f (X)  �  =  �  ∂XcdXba(JT)bc f (X)  �  −  �  {∂XcdXba} (JT)bc f (X)  �  =  �  ∂XcdXba(JT)bc f (X)  �  −  �  χa=d,b=c(JT)bc f (X)  �  =  �  ∂XcdXba(JT)bc f (X)  �  −  �  χa=d(JT)bb f (X)  �  ,  which reduces to zero by application of the fundamental theorem of calculus to the first term  and by making the substitution (JT)bb = Tr JT = 0 in the second term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, for f (X) again a function of the entries of X, observe that  �  ∂(XT)ab + NXba  �  f (X) exp  �  − N  2 Tr(XTX)  �  =  �  ∂(XT)ab f (X)  �  exp  �  − N  2 Tr(XTX)  �  + f (X)  �  ∂(XT)ab exp  �  − N  2 Tr(XTX)  ��  + NXba f (X) exp  �  − N  2 Tr(XTX)  �  =  �  ∂(XT)ab f (X)  �  exp  �  − N  2 Tr(XTX)  �  since ∂(XT)ab exp  �− N  2 Tr(XTX)  �  is equal to −NXba exp  �− N  2 Tr(XTX)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We similarly have  �  ∂Xcd + N(XT)dc  �  f (X) exp  �  − N  2 Tr(XTX)  �  = {∂Xcd f (X)} exp  �  − N  2 Tr(XTX)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) simplifies as  ��  ∂(XT)ab + NXba  �  (JT)bc  �  ∂Xcd + N(XT)dc  �  (Bk1)ad Tr BIn  �  =  ��  ∂(XT)ab + NXba  �  (JT)bc  �  ∂Xcd(Bk1)ad Tr BIn  ��  ,  which can be seen to be precisely the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) upon writing  f (X) = (JT)bc  �  ∂Xcd(Bk1)ad Tr BIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  228  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  Our claim is that using the Leibniz product rule to expand the derivative in the right-hand  side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) and then properly simplifying each of the averages in the resulting  sum will produce our sought loop equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To demonstrate this claim, let us now write the  right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) as  ��  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn  ��  = A +  n  ∑  i=2  �Ai + Ai + Bi  � + ∑  2⩽i,j⩽n,  i̸=j  Cij,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  where  A =  ��  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad  �  Tr BIn  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  Ai =  ��  ∂(XT)ab(Bk1)ad  �  (JT)bc  �  ∂Xcd Tr Bki  �  Tr BIn\\{ki}�  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  Ai =  ��  ∂(XT)ab Tr Bki  �  (JT)bc  �  ∂Xcd(Bk1)ad  �  Tr BIn\\{ki}�  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  Bi =  ��  ∂(XT)ab(JT)bc∂Xcd Tr Bki  �  (Bk1)ad Tr BIn\\{ki}�  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  Cij =  ��  ∂(XT)ab Tr Bki  �  (JT)bc  �  ∂Xcd Tr Bkj  �  (Bk1)ad Tr BIn\\{ki,kj}�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  We show in the following lemma that each of these terms can be expressed in terms of the  mixed moments, so combining equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) will result in a loop equation  on said moments, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us retain the definitions In = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn from  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, for k ∈ N, let [k]mod 2 equal zero when k is even and one when k is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then,  the quantities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) above are given by  A =  ∑  p1+p2=k1−1  mp1,p2,In −  ∑  p1+p2+p3=k1−1  mp1,p2,p3,In + 1 − k2  1  2  mk1−1,In,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  Ai = Ai = 2ki[ki + 1]mod 2  �  mk1+ki−1,In\\{ki} −  ∑  p1+p2=k1−1  mki+p1,p2,In\\{ki}  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  Bi = −2ki[ki + 1]mod 2  �  mk1+ki−1,In\\{ki} +  ∑  p1+p2=ki−1  mk1+p1,p2,In\\{ki}  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  Cij = Cji = −4kikj[ki + 1]mod 2[kj + 1]mod 2 mk1+ki+kj−1,In\\{ki,kj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We present only the computations of A and Cij since their derivations contain all of  the ideas needed to prove equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We first give the proof of equation  229  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26), as it is simpler than that of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we begin by using equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) with (k1, f ) �→ (ki, e) to see that  ∂(XT)ab Tr Bki = ∂(XT)ab(Bki)ee =  ∑  p1+p2=ki−1  �  (JXBki−1)ba + (Bki−1XTJ)ab  �  = (JXBki−1)ba  ∑  p1+p2=ki−1  �  1 + (−1)ki  �  = 2ki[ki + 1]mod 2(JXBki−1)ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  Here, we have used the fact that (Bki−1XTJ)T = JTX(BT)ki−1 = (−1)ki JXBki−1 and that  (1 + (−1)ki) equals two if ki is even and vanishes, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Repeating this argument with  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) likewise shows that  ∂Xcd Tr Bkj = 2kj[kj + 1]mod 2(Bkj−1XTJ)dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Multiplying these two results together, in combination with (JT)bc, then reveals the identity  �  ∂(XT)ab Tr Bki  �  (JT)bc  �  ∂Xcd Tr Bkj  �  = −4kikj[ki + 1]mod 2[kj + 1]mod 2(Bki+kj−1)da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Further multiplying this by (Bk1)ad Tr BIn\\{ki,kj}, summing over the repeated indices a, d, and  then taking the average yields the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Similar to above, we begin the proof of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) by noting that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  with (e, f ) = (a, d) tells us that  (JT)bc∂Xcd(Bk1)ad =  ∑  p1+p2=k1−1  �  (−1)p2(Bk1−1XT)ab − (Bp1XT)ab Tr Bp2  �  = [k1]mod 2(Bk1−1XT)ab −  ∑  p1+p2=k1−1  (Bp1XT)ab Tr Bp2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  where we have used the fact that ∑p1+p2=k1−1(−1)p2 = ∑k1−1  p2=0(−1)p2 equals one if k1 is odd  and zero, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now, taking (k1, e, f ) �→ (k1 − 1, a, b) in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) shows that  ∂(XT)ab(Bk1−1XT)ab =  ∑  p1+p2=k1−2  �  Tr Bp1 Tr Bp2+1 + (−1)p2 Tr Bk1−1�  + N Tr Bk1−1  =  ∑  p1+p2=k1−1  Tr Bp1 Tr Bp2 + [k1 + 1]mod 2 Tr Bk1−1  =  ∑  p1+p2=k1−1  Tr Bp1 Tr Bp2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  the second line follows from making the replacement p2 �→ p2 − 1 in the sum and noting  that the new (p1, p2) = (k1 − 1, 0) term this introduces is equivalent to the term N Tr Bk1−1  230  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  that already appears at the end of the first line, while the final line follows from the fact that  [k1 + 1]mod 2 vanishes whenever k1 is odd, but Tr Bk1−1 vanishes whenever k1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moving  on to the second term in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28), using equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) with the replacement  ki �→ p2 and equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) with k1 �→ p1 shows that  ∂(XT)ab(Bp1XT)ab Tr Bp2 =  �  ∂(XT)ab(Bp1XT)ab  �  Tr Bp2 + (Bp1XT)ab  �  ∂(XT)ab Tr Bp2  �  =  ∑  q1+q2=p1  Tr Bq1 Tr Bq2 Tr Bp2 + [p1]mod 2 Tr Bp1 Tr Bp2 + 2p2[p2 + 1]mod 2 Tr Bp1+p2  =  ∑  q1+q2=p1  Tr Bq1 Tr Bq2 Tr Bp2 + 2p2[p2 + 1]mod 2 Tr Bp1+p2,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  where the last line follows from the fact that [p1]mod 2 vanishes if p1 is even, but Tr Bp1  vanishes if p1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying the operator ∂(XT)ab to both sides of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) and  then substituting in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) shows that  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad = [k1]mod 2  ∑  p1+p2=k1−1  Tr Bp1 Tr Bp2  −  ∑  p1+p2=k1−1  �  2p2[p2 + 1]mod 2 Tr Bp1+p2 +  ∑  q1+q2=p1  Tr Bq1 Tr Bq2 Tr Bp2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  The factor of [k1]mod 2 in the first term can be removed since it is superfluous: for Tr Bp1, Tr Bp2  to be non-zero, each of p1, p2 must be even and, consequently, the sum over p1 + p2 = k1 − 1  is non-zero only if k1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, the second term is non-zero for k1 odd and it  simplifies as  −  ∑  p1+p2=k1−1  2p2[p2 + 1]mod 2 Tr Bp1+p2 = −2 Tr Bk1−1(2 + 4 + · · · + k1 − 1) = 1 − k2  1  2  Tr Bk1−1,  while replacing p1 in the outer sum of the third term by q1 + q2 and then renaming the  summation indices shows that  −  ∑  p1+p2=k1−1  ∑  q1+q2=p1  Tr Bq1 Tr Bq2 Tr Bp2 = −  ∑  q1+q2+p2=k1−1  Tr Bq1 Tr Bq2 Tr Bp2  = −  ∑  p1+p2+p3=k1−1  Tr Bp1 Tr Bp2 Tr Bp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  231  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Combining these observations together, we thus see that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) reduces to  ∂(XT)ab(JT)bc∂Xcd(Bk1)ad =  ∑  p1+p2=k1−1  Tr Bp1 Tr Bp2 + 1 − k2  1  2  Tr Bk1−1  −  ∑  p1+p2+p3=k1−1  Tr Bp1 Tr Bp2 Tr Bp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, multiplying the right-hand side of this equation by Tr BIn and taking the average  produces the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It is now a simple matter of substituting the expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) into the right-  hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) while replacing the left-hand side by N2mk1+1,In, as prescribed  by Propositon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, to obtain the desired loop equation on the mixed moments mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This  loop equation is presented as equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  At this point, it is possible to proceed by substituting the moment-cumulants relation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) into the loop equation on the moments to obtain an analogous equation on the  cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Simplifying this equation on the ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn in an appropriate manner (using a  similar argument to that given in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 upcoming), multiplying  the result by x−k1−1  1  · · x−kn−1  n  , and summing over k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 would then yield the loop  equation on the Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do not go down this path, instead  opting to first derive the loop equation on the unconnected correlators Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  and then transforming it, through the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), into the corresponding loop equation  on the Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) — if necessary, the loop equation on the mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  can be extracted from the loop equation on the Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) by equating coefficients of  x−k1−1  1  · · x−kn−1  n  , possibly by taking suitable residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (Note that we could have alternatively  skipped the derivation of the loop equation on the mixed moments altogether by starting  our analysis at a perturbation of the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), rather than (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' our choice of starting  point leads to the cleaner presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall that Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn), set U0 := 1, and define Un′ := 0 for n′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Furthermore, define the auxiliary function  Ai(x1, Jn) = x2  i (2Un(x1, x1, Jn \\ {xi}) − Un(xi, Jn)) − x1xiUn(x1, Jn)  + 1  x1  �  x1xiUn−1(Jn) − x2  i Un−1(x1, Jn \\ {xi})  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  232  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  Then, for n ⩾ 1, the unconnected correlators specified by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) satisfy the loop equation  0 = x1Un+2(x1, x1, x1, Jn) − Un+1(x1, x1, Jn) + N2x1Un(x1, Jn) − N3Un−1(Jn)  + 1  2x1  �  x2  1  ∂2  ∂x2  1  + x1  ∂  ∂x1  − 1  �  Un(x1, Jn) + 2  n  ∑  i=2  ∂  ∂xi  � Ai(x1, Jn)  x2  1 − x2  i  �  + 8  ∑  2⩽i<j⩽n  1  x1  ∂2  ∂xi∂xj  �  x1x2  i xjUn−2(Jn \\ {xi})  (x2  1 − x2  j )(x2  i − x2  j )  −  x1xix2  j Un−2(Jn \\ {xj})  (x2  1 − x2  i )(x2  i − x2  j )  +  x2  i x2  j Un−2(x1, Jn \\ {xi, xj})  (x2  1 − x2  i )(x2  1 − x2  j )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As mentioned above, inserting the equalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  and combining the result with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) gives the loop equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) on the mixed  moments mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Multiplying both sides of this loop equation by x−k1−1  1  · · x−kn−1  n  and  summing over k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 then produces the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We detail this  transformation term by term in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Loop equations on the ˜W(J1)  n  and W(J1),l  n  Our main goal in this subsection is to transform the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) on the Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  into a loop equation on the Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) through the use of the identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Actually,  we use higher order analogues of this identity, which we present below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall that Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) and let µ ⊢ (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) indicate that µ is a partition  of the ordered set (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', µ = {µt}m  t=1 for some 1 ⩽ m ⩽ n such that the disjoint union  ⊔m  t=1µt = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Writing #µ for the size of µ (m in the preceding sentence), we have that  Un(x1, Jn) =  ∑  K1⊔K2=Jn  U#K1(K1)W#K2+1(x1, K2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  Un+1(x1, x1, Jn) =  ∑  µ⊢(x1,x1)  ∑  ⊔#µ+1  t=1 Kt=Jn  U#K1(K1) ∏  µt∈µ  W#Kt+1+#µt(µt, Kt+1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  Un+2(x1, x1, x1, Jn) =  ∑  µ⊢(x1,x1,x1)  ∑  ⊔#µ+1  t=1 Kt=Jn  U#K1(K1) ∏  µt∈µ  W#Kt+1+#µt(µt, Kt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) is a simple rewriting of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This equation tells us, upon  increasing n by one and replacing Jn with (x1, Jn), that  Un+1(x1, x1, Jn) =  ∑  K1⊔K2=(x1,Jn)  U#K1(K1)W#K2+1(x1, K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  233  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Splitting this sum based on whether or not K1 contains x1 gives  Un+1(x1, x1, Jn) =  ∑  K1⊔K2=Jn  [U#K1+1(x1, K1)W#K2+1(x1, K2) + U#K1(K1)W#K2+2(x1, x2, K2)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Substituting in an appropriate rewriting of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) for U#K1+1(x1, K1) then simplifies  this expression to equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, increasing n by one in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) and  replacing Jn with (x1, Jn) shows that  Un+2(x1, x1, x1, Jn) =  ∑  µ⊢(x1,x1)  ∑  ⊔#µ+1  t=1 Kt=(x1,Jn)  U#K1(K1) ∏  µt∈µ  W#Kt+1+#µt(µt, Kt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Repeating the exercise described above then produces equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Note 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us remark that the notation in the above is drawn from the work [74];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see also  [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We write µ = {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , µ#µ}, which implicitly defines the sets µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , µ#µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Observe  that since we are dealing with partitions of ordered sets, the partition µ = { {x1}, {x1, x1} }  is counted three times: the x1 in µ1 = {x1} could have been the first, second, or third entry  in (x1, x1, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Although the µt must be non-empty, the Kt are allowed to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  With Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 in hand, we can now start computing loop equations for the connected  correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, setting n = 1 in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) and then replacing U0,  U1(x1), U2(x1, x1), and U3(x1, x1, x1) by 1, W1(x1), the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  with n = 1, and the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) with n = 1, respectively, yields the  n = 1 loop equation on the connected correlators,  0 = x1  �  W3(x1, x1, x1) + 3W2(x1, x1)W1(x1) + W1(x1)3� − W2(x1, x1) − W1(x1)2  + N2x1W1(x1) − N3 + 1  2x1  �  x2  1  d2  dx2  1  + x1  d  dx1  − 1  �  W1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  It can be immediately seen that, similar to the loop equations reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, this loop  equation has too many unknowns (W1, W2, and W3) to be solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, in analogy with  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14), substituting the large N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), repeated here for convenience,  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) = N2−n  ∞  ∑  l=0  Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  Nl  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) and equating terms of like order in N produces, for each l ⩾ 0, the  (1, l) loop equation used to compute Wl  1(x1) (recall from the discussion below equation  234  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) that iterating through (n, l) ∈ N2 in the correct order allows us to solve the (n, l)  loop equation for the correlator expansion coefficient Wl  n),  0 = x1  �  Wl−4  3  (x1, x1, x1) + 3  ∑  l1+l2=l−2  Wl1  2 (x1, x1)Wl2  1 (x1) +  ∑  l1+l2+l3=l  Wl1  1 (x1)Wl2  1 (x1)Wl3  1 (x1)  �  − Wl−3  2  (x1, x1) −  ∑  l1+l2=l−1  Wl1  1 (x1)Wl2  1 (x1) + x1Wl  1(x1) − χl=0  + 1  2x1  �  x2  1  d2  dx2  1  + x1  d  dx1  − 1  �  Wl−2  1  (x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39)  (As usual, we have set Wl′  n := 0 for all n ⩾ 1 and l′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') In particular, setting l = 0 in the  above gives the spectral curve  0 = x1  �  W0  1(x1)  �3 + x1W0  1(x1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  Being a cubic polynomial in W0  1(x1), this equation can be solved to express W0  1(x1) as a cube  root (although equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) has three solutions, only one obeys the requirement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  and is thus relevant to our setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do not make W0  1(x1) explicit here, but refer to [127],  [74] and references therein for techniques on obtaining such an expression — in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we  instead find a rational parametrisation for the spectral curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40), in analogy with the  parametrisations x = x(z) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) and W0  1(x) = y(z) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moving on, setting n = 2 in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) and then replacing U1 by W1, U2 by the  right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) with n = 2 and suitable variables, U3(x1, x1, x2) by the  right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) with n = 2, and U4(x1, x1, x1, x2) by the right-hand side  of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) with n = 2 shows that  0 = x1  ∑  µ⊢(x1,x1,x1)  ∑  ⊔#µ+1  t=1 Kt={x2}  U#K1(K1) ∏  µt∈µ  W#Kt+1+#µt(µt, Kt+1)  −  ∑  µ⊢(x1,x1)  ∑  ⊔#µ+1  t=1 Kt={x2}  U#K1(K1) ∏  µt∈µ  W#Kt+1+#µt(µt, Kt+1) − N3U1(x2)  +  �  N2x1 + 1  2x1  �  x2  1  ∂2  ∂x2  1  + x1  ∂  ∂x1  − 1  ��  (W2(x1, x2) + W1(x1)U1(x2))  + 2  x1  ∂  ∂x2  1  x2  1 − x2  2  �  x1x2  2  �  2W2(x1, x1) + 2W1(x1)2 − W2(x2, x2) − W1(x2)2�  − x2  1x2 (W2(x1, x2) + W1(x1)W1(x2))  + x1x2W1(x2) − x2  2W1(x1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41)  235  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  This is not yet our sought loop equation, as we can simplify it one step further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed,  subtracting the product of the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) by U1(x2), which is zero,  from equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) reveals the n = 2 loop equation on the connected correlators,  0 = x1  �  W4(x1, x1, x1, x2) + 3W3(x1, x1, x2)W1(x1)  + 3W2(x1, x1)W2(x1, x2) + 3W2(x1, x2)W1(x1)2�  − W3(x1, x1, x2)  − 2W2(x1, x2)W1(x1) + N2x1W2(x1, x2) + 1  2x1  �  x2  1  ∂2  ∂x2  1  + x1  ∂  ∂x1  − 1  �  W2(x1, x2)  + 2  x1  ∂  ∂x2  1  x2  1 − x2  2  �  x1x2  2  �  2W2(x1, x1) + 2W1(x1)2 − W2(x2, x2) − W1(x2)2�  − x2  1x2 (W2(x1, x2) + W1(x1)W1(x2)) + x1x2W1(x2) − x2  2W1(x1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42)  As with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37), equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) is unsolvable, but substituting in the large  N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) results in a set of solvable loop equations on the correlator expansion  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we have for each l ⩾ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' l) loop equation used to compute the  expansion coefficient Wl  2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  0 = x1  �  Wl−4  4  (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2) + 3  ∑  l1+l2=l−2  Wl1  3 (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2)Wl2  1 (x1)  + 3  ∑  l1+l2=l−2  Wl1  2 (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1)Wl2  2 (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2) + 3  ∑  l1+l2+l3=l  Wl1  2 (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2)Wl2  1 (x1)Wl3  1 (x1)  �  − Wl−3  3  (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2) − 2  ∑  l1+l2=l−1  Wl1  2 (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2)Wl2  1 (x1) + x1Wl  2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2)  + 1  2x1  �  x2  1  ∂2  ∂x2  1  + x1  ∂  ∂x1  − 1  �  Wl−2  2  (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2)  + 2  x1  ∂  ∂x2  1  x2  1 − x2  2  �  2x1x2  2Wl−2  2  (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x1) − x1x2  2Wl−2  2  (x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2) − x2  1x2Wl−2  2  (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' x2)  + x1x2  2 ∑  l1+l2=l  �  2Wl1  1 (x1)Wl2  1 (x1) − Wl1  1 (x2)Wl2  1 (x2)  �  + x1x2Wl−1  1  (x2)  − x2  1x2 ∑  l1+l2=l  Wl1  1 (x1)Wl2  1 (x2) − x2  2Wl−1  1  (x1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43)  Setting l = 0 then specifies W0  2(x1, x2) as a rational function of W0  1 and its derivative:  W0  2(x1, x2) = 2  x1  1  3(W0  1(x1))2 + 1  ∂  ∂x2  x2  x2  2 − x2  1  �  2x2(W0  1(x1))2 − x2(W0  1(x2))2  − x1W0  1(x1)W0  1(x2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  236  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  We solve this equation using the rational parametrisation mentioned below equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As expected, we will recover the structure of the Bergman kernel (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) and the discussion below it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Let us now highlight that in obtaining equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) from equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41), we have  simply subtracted off terms from the first three lines of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) that contain a factor  of U1(x2) while using equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37) to argue that this does not change the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  This idea extends to the general n ⩾ 3 case:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us retain the notation used in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For n ⩾ 3, the connected correlators  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) of the global scaled (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N) antisymmetrised Laguerre ensemble satisfy the loop equation  0 =  �  �x1  ∑  µ⊢(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x1)  −  ∑  µ⊢(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x1)  �  �  ∑  ⊔#µ  t=1Kt=Jn  ∏  µt∈µ  W#Kt+#µt(µt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Kt)  + N2x1Wn(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Jn) + 1  2x1  �  x2  1  ∂2  ∂x2  1  + x1  ∂  ∂x1  − 1  �  Wn(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Jn)  + 2  x1  n  ∑  i=2  ∂  ∂xi  1  x2  1 − x2  i  ��  2x1x2  i  ∑  µ⊢(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='x1)  −x1x2  i  ∑  µ⊢(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='xi)  −x2  1xi  ∑  µ⊢(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='xi)  �  ×  ∑  ⊔#µ  t=1Kt=Jn\\{xi} ∏  µt∈µ  W#Kt+#µt(µt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Kt)  + x1xiWn−1(Jn) − x2  i Wn−1(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Jn \\ {xi})  �  + 8  ∑  2⩽i<j⩽n  1  x1  ∂2  ∂xi∂xj  �  x1x2  i xjWn−2(Jn \\ {xi})  (x2  1 − x2  j )(x2  i − x2  j )  −  x1xix2  j Wn−2(Jn \\ {xj})  (x2  1 − x2  i )(x2  i − x2  j )  +  x2  i x2  j Wn−2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Jn \\ {xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' xj})  (x2  1 − x2  i )(x2  1 − x2  j )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We give a modification of the inductive argument detailed in [142, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A] (see also  [74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To proceed, let us define E1(x1) to be the right-hand side of the n = 1 loop equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37), E2(x1, x2) to be the right-hand side of the n = 2 loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42), and for n ⩾ 3,  define En(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) to be the right-hand side of the alleged loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Our induction hypothesis is that Em(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xm) = 0 for all 1 ⩽ m ⩽ n — this has already  been shown through equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='42) to be true for the base cases m = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In a  similar manner to the derivation of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41), use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 to write the Un loop  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) in terms of the Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, check term by term that the right-hand side of  237  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  this equation agrees with the sum  ∑  K1⊔K2=Jn  U#K1(K1)E#K2+1(x1, K2) = En(x1, Jn) +  ∑  K1⊔K2=Jn,  K1̸=∅  U#K1(K1)E#K2+1(x1, K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Since the left-hand side of the equation obtained from using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 to rewrite the Un  loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) is zero, the above sum must also vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, we have  0 = En(x1, Jn) +  ∑  K1⊔K2=Jn,  K1̸=∅  U#K1(K1)E#K2+1(x1, K2) = En(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn),  where the second equality follows from our induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As En(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) is the  right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Like with the loop equations given earlier in this subsection, to extract information from  the above loop equation, one must substitute the large N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) into said equation  and collect terms of equal order in N to obtain the (n, l) loop equations on the correlator  expansion coefficients Wl  n — it is the (n, l) loop equations that can be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We do not  display the Wl  n loop equations here since it is straightforward to extract them from equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34): one need only replace products of the form Wn1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn1) · · · Wnk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xnk) with  sums of the form  ∑  l1+···+lk=l−r  Wl1n1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn1) · · · Wlknk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xnk),  where r depends on what power of N the original product is multiplied by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' we refer to the  derivations of equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='43) for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As for the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) itself, some brief comparisons to results in the existing  literature are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First, let us observe that for each n ⩾ 3, the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) is  indeed higher order than the equivalent loop equations seen in the classical and general β  ensembles reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' By this, we mean that the equivalent of the term Wn+1(x1, x1, Jn)  in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8) is the term Wn+2(x1, x1, x1, Jn) corresponding to the µ = { {x1, x1, x1} }  term in first sum displayed in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In fact, the fact that this sum is over  µ ⊢ (x1, x1, x1), as opposed to µ ⊢ (x1, x1), classifies this loop equation as ‘higher order’ than  in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Another way of quantifying the complexity of our loop equation is to  note that we have sums over partitions of various size-two ordered sets appearing in many  of the terms and our largest sum contains products of three connected correlators — this  238  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  is not seen in more traditional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, if we were to simplify the Wl  n loop equation  corresponding to the Wn loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) using residue calculus in a similar manner to  that presented in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, our analogue of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41), if it exists, would need at least an  extra sum of a product of three differential forms ωl′  n′ on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moving past comparisons to loop equations for β ensembles, let us recall that even  though the above observations show that the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) is structurally quite  different from most of the loop equations studied in the literature, there still exist some  works studying such higher order loop equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Namely, multi-matrix models and chains  of matrices have been shown to be characterised by such higher order loop equations [110],  [268], while loop equations involving large sums of the form ∑µ⊢(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',x1) have been studied  in the abstract setting for their own sake [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We also remind the reader that loop  equations for the (N, N, N) complex Wishart product ensemble were derived in [74], where  they were shown to be of the same order as equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Nonetheless, we highlight  the fact that the antisymmetric nature of the matrix B = ˜J1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='60) shows itself in the loop  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) in a way that has not been seen in previously studied loop equations:  Comparing the last line of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) to the third line of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), we see that  the denominator x1 − xi has been replaced by x2  1 − x2  i , which is unchanged when mapping  x1 �→ −x1 or xi �→ −xi (this structure is also seen in the last two lines of our loop equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It can be surmised from the computations of Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 that these differences of squares  arise from the fact that the mixed moments mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn vanish whenever any of the ki are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Discussion on W(J1),0  1  and W(J1),0  2  Let us now focus our discussion on the (1, 0) and (2, 0) loop equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44)  so that we may comment on the large N limiting behaviour of the global scaled (N, N)  antisymmetrised Laguerre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Taking n, l to be such small integers also makes  it feasible for us to round out this section with some concrete computations that make  connections with the content of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We commence our discussion by noting that W0  1(x1) has the large x1 expansion  W0  1(x1) = W(J1),0  1  (x1) =  ∞  ∑  k=0  ˜M(J1)  k,0  xk+1  1  ,  239  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  in keeping with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and having adopted the notation of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) to  write ˜m(J1)  k  = ∑∞  l=0 ˜M(J1)  k,l  N−l (as allowed by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting ˜M(iJ1)  k,0  denote the  analogous scaled moments of the matrix iJ1, we have that  ˜M(J1)  k,0  = lim  N→∞  �  Tr  ˜J k  1  �  = (−i)k lim  N→∞  �  Tr (i ˜J1)k�  = (−i)k ˜M(iJ1)  k,0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46)  As these latter moments are also characterised as the spectral moments of the large N = N0  limit ρ(iJ1),0(λ) of the scaled density ˜ρ(iJ1)(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20) specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13 of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1,  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) tells us that  ˜M(J1)  2k,0 = (−1)km(FC2)  k  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  where m(FC2)  k  is the m = 2 Fuss–Catalan number specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) (recall that  ˜M(J1)  k,0  vanishes for odd values of k due to the fact that J1 is antisymmetric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, one can  check using computer algebra that the solution of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) has large x1 expansion  W0  1(x1) = 1  x1  − 1  x3  1  + 3  x5  1  − 12  x7  1  + 55  x9  1  − 273  x11  1  + O(x−13  1  );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48)  observe that for integer k, the coefficient of x−2k−1  1  is precisely (−1)km(FC2)  k  = (−1)k  1  2k+1(3k  k ),  while there are no terms of the form x−2k  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Rewriting equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='46) in terms of the moment generating functions defined by  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) and using equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47), we see that  W(J1),0  1  (x1) =  ∞  ∑  k=0  ˜M(J1)  2k,0  x2k+1  1  = −x1  ∞  ∑  k=0  m(FC2)  k  (−x2  1)k+1 = −x1W(FC2)  1  (−x2  1),  where W(FC2)  1  (x) := ∑∞  k=0 m(FC2)  k  /xk+1 is the generating function of the relevant Fuss–Catalan  numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equivalently, it is the Stieltjes transform (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) of the m = 2 Fuss–Catalan  distribution ρ(FC2)(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In combination with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40), this suggests that upon  writing u = −x2  1, the resolvent W(FC2)  1  (u) satisfies the functional relation  0 = u2 �  W(FC2)  1  (u)  �3  − uW(FC2)  1  (u) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49)  This is known to be true (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [127], whence equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) could be obtained), and  thus serves as confirmation that the loop equation analysis presented in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 has produced  the correct spectral curve for the global scaled (N, N) antisymmetrised Laguerre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  240  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For another consistency check of our spectral curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40), we can compute it  using techniques from free probability theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' [244]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, such techniques  are designed to probe the macroscopic statistics of products of random matrices and are  thus ill suited for accessing statistics that are not of leading order in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These lower order  statistics are of interest to us since, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, our 1/N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) has  combinatorial meaning at all orders in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Moreover, these combinatorial interpretations are  valid for finite N, while free probability theory is concerned with the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us  thus draw attention to the fact that our loop equation analysis allows us to compute the  mixed cumulants ˜c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to any desired order in N, assuming the availability of enough  computation power and time (recall the discussion below equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In keeping with the discussion at the end of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, let us reiterate that, as foreshadowed  at the beginning of this chapter, our spectral curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) is a third order polynomial in  W0  1(x1) = W(J1),0  1  (x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, finding a rational parametrisation of our spectral curve is not  as straightforward as, say, in the case of the one-cut 1-Hermitian matrix model reviewed  in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However, we are quite fortuitous in that the spectral curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) is a linear  polynomial in x1, so we may rearrange it to write  x1 =  1  �  W0  1(x1)  �3 + W0  1(x1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Thus, in analogy with the parametrisations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), a natural parametrisation of the  above equation presents itself to us: Simply define  x(z) =  1  z3 + z,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50)  y(z) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)  Then, replacing x1 by x(z1) and W0  1(x1) = W0  1(x(z1)) by its analytic continuation y(z1) in the  right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40) returns zero, showing that this is a valid parametrisation  of our spectral curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using the Newton polygon method (see [32] and references therein)  on the complex algebraic curve 0 = xy3 + xy − 1 shows that our spectral curve is the genus  zero Riemann sphere with viable coordinates x(z), y(z) as given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, W0  1(x(z1)) is  a single-valued function of z1 ∈ CP1 whose value agrees with W0  1(x1) when z1 ∈ x−1(x1)  is chosen to be close to z1 = 0 (this ensures that we choose the solution of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='40)  obeying the asymptotic requirement (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) that W0  1(x1) ∼ 1/x1 as x1 → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  241  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Now, taking z1, z2 ∈ CP1 and substituting x1 = x(z1) and x2 = x(z2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50) into  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='44) shows that  W0  2(x(z1), x(z2)) =  1  x′(z1)x′(z2)  1  (z1 − z2)2 −  1  (x(z1) − x(z2))2  −  1  x′(z1)x′(z2)  1  (z1 + z2)2 + (x(z1) + x(z2))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  The first line of the right-hand side is the value of W0  2(x(z1), x(z2)) obtained in the case of  the one-cut 1-Hermitian matrix model upon rearranging equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34), while the second  line ensures that W0  2(x(z1), x(z2)) is antisymmetric in z1, z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we have a double pole  at z1 = −z2 in addition to the double pole at z1 = z2 seen in the ensembles reviewed in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This phenomenon is consistent with the appearance of the denominators of  the form x2  a − x2  b in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45), which was briefly discussed at the end of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In  keeping with Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 above, we show that our loop equation analysis can determine  W1  1(x1), which is the 1/N correction to W0  1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is a simple matter of setting l = 1 in  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='39) with x1 replaced by x(z1) and W0  1(x1) replaced by y(z1) = z1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51) to see  that  W1  1(x(z1)) =  z2  1  x(z1)(3z2  1 + 1) = z5  1 + z3  1  3z2  1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53)  From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, we know that the mixed cumulants ˜c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn have a genus expansion  of the form  ˜c(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = N2−n  k1+···+kn+1−n  ∑  l=0  c(J1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  Nl  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54)  Combining this with the large N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) and large x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) of  the connected correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) then shows that  Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  ∞  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn=0  c(J1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  xk1+1  1  · · xkn+1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='55)  Hence, noting that these interpretations as generating functions are valid only in the large  x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn regime, we have that the coefficients of the cumulant genus expansions can be  extracted from the correlator expansion coefficients using the residue formula  c(J1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = (−1)n  Res  x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',xn=∞xk1  1 · · · xkn  n Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) dx1 · · · dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56)  As our formalism produces expressions for the correlator expansion coefficients in terms of  the variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , zn ∈ CP1 through the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50), this residue formula is only useful  242  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Antisymmetrised Laguerre Ensemble  to us upon reformulating it in terms of the z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , zn variables, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, observe that  xi = x(zi) = ∞ corresponds to zi = 0, ±i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Of these three solutions, we are interested in  zi = 0, as it coincides with the requirement that y(zi) = zi → 0 as x(zi) → ∞, which follows  from the fact that the branch of W0  1(x1) that behaves as a moment generating function is  the one that exhibits the asymptotic W0  1(x1) ∼ 1/x1 in the x1 → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Noting also that  setting xi = x(zi) means that dxi = x′(zi)dzi, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56) therefore implies that  c(J1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = (−1)n  Res  z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',zn=0Wl  n(x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn))  n  ∏  i=1  x(zi)kix′(zi) dzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57)  For example, inserting W0  1(x(z1)) = z1 into the above shows that c(J1),0  4  = 3, which  is exactly the coefficient of 1/x5  1 in the large x1 expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, substituting  equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) into the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57) shows that c(J1),1  2  = 1 and c(J1),0  2,2  = 12  — see Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 for complementary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, these computations  imply that, up to the topological equivalence relation described above Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15, there  are three distinct genus zero rooted topological hypermaps with a degree four black vertex,  one or two red vertices of even valency, and an even number of twisted edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' there is one  rooted topological hypermap of Euler genus one with one bivalent black vertex, one bivalent  red vertex, and one twisted edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and there are twelve genus zero 2-rooted topological  hypermaps with two degree two black vertices, one or two red vertices of even valency,  and an even number of twisted edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The one topological hypermap counted by c(J1),1  2  has already been displayed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, while the fifteen topological hypermaps counted  by c(J1),0  4  and c(J1),0  2,2  are shown in Figures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that in general, |c(J1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn|  may take a value which is less than the number of hypermaps it corresponds to, due to the  cancelling in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='65) — we do not see this here due to n, l being too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3: These three topological hypermaps are counted by c(J1),0  4  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The root edges  are coloured blue with notches indicating local orientation of the black vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' All the  edges are untwisted, so we do not need the ±1 labelling discussed in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  243  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4: These twelve topological hypermaps are counted by c(J1),0  2,2  = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have again  coloured the two root edges blue, distinguished them with the labels 1, 2, and used notches  to mark the local orientation of each black vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We have avoided drawing complicated  deformations of the sphere, like in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18, by having all our edges be untwisted and  instead letting our black vertices have possibly inconsistent local orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Loop Equations for the Hermitised Laguerre Ensemble  This section is concerned with the derivation of loop equations for the global scaled (N, N)  Hermitised Laguerre ensemble defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In parallel to the development  of the previous section, we derive loop equations for the unconnected n-point correlators  ˜U(H1)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and the connected n-point correlators ˜W(H1)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), along with the coefficients W(H1),l  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) of their large N expansions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), in  §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These loop equations are used to compute W(H1),0  1  (x1) and W(H1),0  2  (x1, x2) in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  In contrast to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, the discussion of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1–§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 will be relatively succinct: we focus  on presenting analogues of the results given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, providing technical details of  244  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Hermitised Laguerre Ensemble  their derivations only when they differ to those shown earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' As one might expect, the  mechanisms underlying the results of Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To begin, let us redefine X to be the scaled complex N × N Ginibre matrix with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  ˜P(GinUE)(X) =  � N  π  �N2  exp  �  −N Tr(X†X)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  and B to be the product B := X† ˜HX, where ˜H is the scaled N × N GUE matrix with p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  ˜P(GUE)( ˜H) = 2N(N−1)/2  � N  2π  �N2/2  exp  �− N  2 Tr( ˜H2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  Note that X = ˜G in distribution, with ˜G = ˜G1 as in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18, ˜H is as in  said propositions (this matrix represents a slightly different global scaling of the GUE to the  scaling specified in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), and therefore B = ˜H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With regards to the quantities  of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10, equivalently equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), we have that ˜H = √  2/NH, X = G1/  √  N,  and B =  √  2N−3/2H1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting ⟨ · ⟩ now denote averages with respect to  the product ˜P(GinUE)(X) ˜P(GUE)( ˜H), we again simplify notation by omitting tildes and the  superscript (H1) to write  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  �  n  ∏  i=1  Tr Bki  �  ,  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ∈ N,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  ∞  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn=0  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  xk1+1  1  · · xkn+1  n  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  as in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) — recall from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 that mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = 0 when k1 + · · · + kn is  odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The associated mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and connected correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) are  specified by equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29), respectively (we also recall equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Compared to the statistics of the unscaled matrix H1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1), indicated by the superscript  (H1), but without any tildes, the statistical equivalence B =  √  2N−3/2H1 implies that  mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  � √  2  N3/2  �k1+···+kn  m(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn =  � √  2  N3/2  �k1+···+kn  c(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  � N3/2  √  2  �n  U(H1)  n  (N3/2x1/  √  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N3/2xn/  √  2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  � N3/2  √  2  �n  W(H1)  n  (N3/2x1/  √  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , N3/2xn/  √  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  245  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  As in the previous section, the mixed cumulants ck1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn having the genus expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63)  means that Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) admits the large N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We retain our convention  of denoting the associated expansion coefficients as Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) and note that, by the  discussion following Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16, Wl  n = 0 whenever l is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Emulating the key idea of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, we commence our analysis of the global scaled  (N, N) Hermitised Laguerre ensemble by considering the average  ��  ∂(X†)ab + NXba  � �  ∂( ˜H†)bc + N ˜Hcb  � �  ∂Xcd + N(X†)dc  �  (Bk1)ad Tr BIn  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  in analogy with the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) (In and Tr BIn are as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This choice of  starting point is inspired by the following computations:  ∂(X†)ab exp  �  −N Tr(X†X)  �  = −N  �  ∂(X†)ab  N  ∑  e,f =1  (X†)e f Xf e  �  exp  �  −N Tr(X†X)  �  = −N  N  ∑  e,f =1  � �  ∂(X†)ab(X†)e f  �  Xf e  + (X†)e f  �  ∂(X†)abXf e  � �  exp  �  −N Tr(X†X)  �  = −N  N  ∑  e,f =1  �  χe=a,f =bXf e + 0  �  exp  �  −N Tr(X†X)  �  = −NXba exp  �  −N Tr(X†X)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  ∂( ˜H†)bc exp  �− N  2 Tr( ˜H2)  � = − N  2  N  ∑  e,f =1  � �  ∂( ˜H†)bc ˜He f  �  ˜Hf e  + ˜He f  �  ∂( ˜H†)bc ˜Hf e  � �  exp  �− N  2 Tr( ˜H2)  �  = −N ˜Hcb exp  �− N  2 Tr( ˜H2)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  ∂Xcd exp  �  −N Tr(X†X)  �  = −N(X†)dc exp  �  −N Tr(X†X)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  The first two equalities in the above follow from the chain and Leibniz product rules and  the third equality is due to the fact that, as a complex variable, Xf e is independent of  (X†)ab = Xba for any choice of e, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Similar reasoning yields equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  upon recalling that ˜H being Hermitian implies that ˜H† = ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, let us now proceed by computing the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) in two different  ways, thereby deriving loop equations on the mixed moments mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and, consequently, the  unconnected correlators Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  246  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Hermitised Laguerre Ensemble  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Loop equations on the ˜U(H1)  n  Since (X†)ab = Xba and Xcd are independent of each other for all choices of a, b, c, d, the  analogue of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 relevant to the present setting is comparatively simple:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Having set B = X† ˜HX as above, the partial derivatives of (Bk1)e f with respect to  (X†)ab, ˜Hbc, and Xcd are respectively given by  ∂(X†)ab(Bk1)e f =  ∑  p1+p2=k1−1  (Bp1)ea( ˜HXBp2)b f ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  ∂ ˜Hbc(Bk1)e f =  ∑  p1+p2=k1−1  (Bp1X†)eb(XBp2)c f ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  ∂Xcd(Bk1)e f =  ∑  p1+p2=k1−1  (Bp1X† ˜H)ec(Bp2)d f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  Similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, these sums are empty when k1 = 0 and are otherwise taken over the range  p1 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , k1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The indices e, f are arbitrary and the above equations hold true upon setting  e, f equal to each other or to one of a, b, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moving on, the analogue of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 corresponding to the global scaled (N, N)  Hermitised Laguerre ensemble is as follows:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Writing In = (k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn as usual, the average  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) simplifies to both the left- and right-hand sides of the equation  ��  ∂(X†)ab∂( ˜H†)bc∂Xcd(Bk1)ad Tr BIn  ��  = N3mk1+1,In − k1  ∑  p1+p2=k1−1  mp1,p2,In − 2  ∑  2⩽i<j⩽n  kikjmk1,ki+kj−1,In\\{ki,kj}  −  n  ∑  i=2  ki  �  2k1mk1+ki−1,In\\{ki} +  ∑  p1+p2=ki−1  mk1,p1,p2,In\\{ki}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Expanding the first two brackets in the argument of the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) and using the  independence of ˜H from X†, X shows that  ��  ∂(X†)ab + NXba  � �  ∂( ˜H†)bc + N ˜Hcb  � �  ∂Xcd + N(X†)dc  �  (Bk1)ad Tr BIn  �  =  �  ∂(X†)ab  �  ∂( ˜H†)bc + N ˜Hcb  � �  ∂Xcd + N(X†)dc  �  (Bk1)ad Tr BIn  �  + N  �  ∂( ˜H†)bcXba  �  ∂Xcd + N(X†)dc  �  (Bk1)ad Tr BIn  �  + N2 �  Xba ˜Hcb  �  ∂Xcd + N(X†)dc  �  (Bk1)ad Tr BIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  247  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  The first two averages on the right-hand side vanish by the fundamental theorem of calculus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  while the third average can be simplified using integration by parts:  N2 �  Xba ˜Hcb  �  ∂Xcd + N(X†)dc  �  (Bk1)ad Tr BIn  �  = N3 �  Tr Bk1+1 Tr BIn  �  + N2 �  Xba ˜Hcb∂Xcd(Bk1)ad Tr BIn  �  = N3 �  Tr Bk1+1 Tr BIn  �  + N2 �  ∂XcdXba ˜Hcb(Bk1)ad Tr BIn  �  − N2 �  {∂XcdXba} ˜Hcb(Bk1)ad Tr BIn  �  = N3 �  Tr Bk1+1 Tr BIn  �  − N2 �  χc=b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='d=a ˜Hcb(Bk1)ad Tr BIn  �  = N3mk1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In − N2 �  Tr ˜H Tr Bk1 Tr BIn  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  the average  �  ∂XcdXba ˜Hcb(Bk1)ad Tr BIn�  in the third line vanishes due to the fundamental  theorem of calculus, yet again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Now, the average  �  Tr ˜H Tr Bk1 Tr BIn�  poses a challenge not  seen in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, wherein the analogous average  �  Tr J Tr Bk1 Tr BIn�  was  zero due to the antisymmetry of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Here, we must derive auxiliary identities to deal with  this average, in a similar fashion to how identities resulting from applying integration by  parts to the averages (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) were needed to derive loop equations for the Jacobi  β ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The fundamental theorem of calculus (left-hand side) and the Leibniz product  rule combined with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 (right-hand side) shows that  0 =  �  ∂( ˜H†)aa Tr Bk1 Tr BIn  �  = k1  �  Tr(Bk1−1X†X) Tr BIn  �  +  n  ∑  i=2  ki  �  Tr(Bki−1X†X) Tr Bk1 Tr BIn\\{ki}�  − N  �  Tr ˜H Tr Bk1 Tr BIn  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  The averages involving traces of the form Tr(Bkj−1X†X) can be expressed in terms of the  mixed moments by likewise showing that for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n,  0 =  �  ∂(X†)ab(Bki−1X†)ab Tr B{k1}∪In\\{ki}�  = −N  �  Tr(Bki−1X†X) Tr B{k1}∪In\\{ki}�  +  ∑  p1+p2=ki−1  mp1,p2,{k1}∪In\\{ki}  + ∑  1⩽j⩽n,  j̸=i  kjmki+kj−1,{k1}∪In\\{ki,kj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  248  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Hermitised Laguerre Ensemble  Setting i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n into this equation and substituting the resulting set of identities into  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) multiplied by N shows that  −N2 �  Tr ˜H Tr Bk1 Tr BIn  �  = −Nk1  �  Tr(Bk1−1X†X) Tr BIn  �  − N  n  ∑  i=2  ki  �  Tr(Bki−1X†X) Tr Bk1 Tr BIn\\{ki}�  = −k1  ∑  p1+p2=k1−1  mp1,p2,In − k1  n  ∑  j=2  kjmk1+kj−1,In\\{kj} −  n  ∑  i=2  ki  ∑  p1+p2=ki−1  mk1,p1,p2,In\\{ki}  −  n  ∑  i=2  ki  �  �  �  �k1mk1+ki−1,In\\{ki} + ∑  2⩽j⩽n,  j̸=i  kjmk1,ki+kj−1,In\\{ki,kj}  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Substituting this result into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) then gives the right-hand side of equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To show that the average (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) simplifies to the left-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), one  need only repeat the relevant arguments given in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, with the  identities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The next step in obtaining loop equations on the mixed moments is to expand the  right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) using the Leibniz product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting  fi(B) =  �  �  �  �  �  (Bk1)ad,  i = 1,  Tr Bki,  i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , n  and writing Ai,j,h for the average obtained by replacing fi(B) by {∂(X†)ab fi(B)}, then fj(B) by  {∂( ˜H†)bc fj(B)}, and finally fh(B) by {∂Xcd fh(B)} in ⟨ f1(B) · · · fn(B)⟩ so that, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',  A3,3,1 =  ��  ∂(X†)ab∂( ˜H†)bc Tr Bk3  � �  ∂Xcd(Bk1)ad  �  Tr BIn\\{k3}�  ,  we have that the left-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) is given by  ��  ∂(X†)ab∂( ˜H†)bc∂Xcd(Bk1)ad Tr BIn  ��  =  n  ∑  i,j,h=1  Ai,j,h  = A1,1,1 +  n  ∑  i=2  (A1,1,i + A1,i,1 + Ai,1,1 + A1,i,i + Ai,1,i + Ai,i,1 + Ai,i,i)  + ∑  2⩽i,j⩽n,  i̸=j  �A1,i,j + Ai,1,j + Ai,j,1 + Ai,i,j + Ai,j,i + Aj,i,i  � +  ∑  2⩽i,j,h⩽n,  i̸=j̸=h̸=i  Ai,j,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  249  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Every term on the right-hand side of this equation can be expressed in terms of the mixed  moments using similar methods to those detailed in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5  supplanting the role of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, substituting this equation into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  produces the aforementioned loop equation on the mixed moments — this loop equation  and the required expressions for the Ai′,j′,h′ are displayed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Multiplying both  sides of said loop equation by x−k1−1  1  · · x−kn−1  n  and summing over k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 then gives  the following loop equation on the unconnected correlators — this transformation is carried  out term by term in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall that Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn), set U0 := 1, and define Un′ := 0 for n′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Furthermore, define the auxiliary functions (not to be confused with that given in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  Ai(x1, Jn) = 3x1xiUn+1(x1, x1, x1, Jn \\ {xi}) − x1xiUn+1(x1, x1, Jn)  x1 − xi  − x1xiUn+1(x1, xi, Jn) + x2  i Un+1(xi, xi, Jn)  x1 − xi  +  �3x1  2  ∂2  ∂x2  1  +  ∂  ∂x1  + xi  ∂2  ∂x1∂xi  + xi  2x1  ∂2  ∂x2  i  xi  �  ×  � xiUn−1(x1, Jn \\ {xi}) − x1Un−1(Jn)  x1 − xi  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  Ai,j(x1, Jn) = 2xiUn−1(x1, Jn \\ {xi}) − 2xjUn−1(x1, Jn \\ {xj})  xi − xj  + xixj  �  6Un−1(x1, x1, Jn \\ {xi, xj}) + Un−1(Jn)  (x1 − xi)(x1 − xj)  − 3Un−1(x1, Jn \\ {xj}) + 3Un−1(x1, , Jn \\ {xi})  (x1 − xi)(x1 − xj)  − 3Un−1(xi, Jn \\ {xj}) − 2Un−1(x1, Jn \\ {xi})  (x1 − xi)(xi − xj)  + 3Un−1(xj, Jn \\ {xi}) − 2Un−1(x1, Jn \\ {xj})  (x1 − xj)(xi − xj)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  Ai,j,h(x1, Jn) = xixjxhUn−3(x1, Jn \\ {xi, xj, xh})  (x1 − xi)(x1 − xj)(x1 − xh)  − x1xjxhUn−3(Jn \\ {xj, xh})  (x1 − xi)(xi − xj)(xi − xh)  + x1xixhUn−3(Jn \\ {xi, xh})  (x1 − xj)(xi − xj)(xj − xh) −  x1xixjUn−3(Jn \\ {xi, xj})  (x1 − xh)(xi − xh)(xj − xh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  250  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Hermitised Laguerre Ensemble  Then, for n ⩾ 1, the unconnected correlators specified by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) satisfy the loop equation  0 = x2  1Un+3(x1, x1, x1, x1, Jn) − N3x1Un(x1, Jn) + N4Un−1(Jn)  + lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  � �  Un+1(ξ, x1, Jn) + 1  2Un+1(x1, x1, Jn)  �  +  n  ∑  i=2  ∂  ∂xi  Ai(x1, Jn)  +  ∑  2⩽i<j⩽n  ∂2  ∂xi∂xj  Ai,j(x1, Jn) + 6  x1  ∑  2⩽i<j<h⩽n  ∂3  ∂xi∂xj∂xh  Ai,j,h(x1, Jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Loop equations on the ˜W(H1)  n  and W(H1),l  n  Setting n = 1 in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) above and using the canonical extension of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  specifying Un+3(x1, x1, x1, x1, Jn) as a suitable sum over µ ⊢ (x1, x1, x1, x1) produces the n = 1  loop equation on the connected correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29),  0 = x2  1  ∑  µ⊢(x1,x1,x1,x1) ∏  µt∈µ  W#µt(µt) − N3x1W1(x1) + N4  + lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  � �  W2(ξ, x1) + 1  2W2(x1, x1)  + W1(ξ)W1(x1) + 1  2 (W1(x1))2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  Inserting the 1/N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) into this equation and collecting terms of order N4−l  (l ⩾ 0) yields the (1, l) loop equation  0 = x2  1  ∑  µ⊢(x1,x1,x1,x1)  ∑  l1+···+l#µ  =l+2#µ−8  ∏  µt∈µ  Wlt  #µt(µt) − x1Wl  1(x1) + χl=0  + lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  � �  Wl−4  2  (ξ, x1) + 1  2Wl−4  2  (x1, x1)  +  ∑  l1+l2=l−2  Wl1  1 (x1)  �  Wl2  1 (ξ) + 1  2Wl2  1 (x1)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  (Recall our convention of setting Wl′  n := 0 for all n ⩾ 1 and l′ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=') Setting l = 0 in the above  gives the spectral curve  0 = x2  1  �  W0  1(x1)  �4 − x1W0  1(x1) + 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  while setting l = 1 shows that  0 =  �  4x1  �  W0  1(x1)  �3 − 1  �  W1  1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  251  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) has been given in [143], where it was derived through techniques from  free probability theory and also through the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) between the limiting density  ρ(Hm),0(λ) of the Hermitised matrix product ensemble (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) and the Fuss–  Catalan distribution ρ(FC2m+1)(λ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) — we have that W(Hm),0  1  (x1) = x1W(FC2m+1)  1  (x2  1) [143]  and that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='49) is known [127] to have the generalisation  0 = um �  W(FCm)  1  (u)  �m+1  − uW(FCm)  1  (u) + 1,  m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28)  Since 4x1  �  W0  1(x1)  �3 = 1 is in contradiction with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26), equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) tells us  that W1  1(x1) must be identically zero — this is perfectly in line with our earlier observation  (see below Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) that Wl  n = 0 whenever l is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For notational convenience, define Ai(x1, Jn) to be Ai(x1, Jn), as specified  by equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20), but with Un−1(x1, Jn \\ {xi}) replaced by Wn−1(x1, Jn \\ {xi}), Un−1(Jn)  replaced by Wn−1(Jn), and each instance of Un+1(K) (K an ordered set of variables drawn  from {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn}) replaced by  ∑  µ⊢K′  ∑  ⊔#µ  t=1Kt=Jn\\{xi} ∏  µt∈µ  W#Kt+#µt(µt, Kt),  where K′ is the ordered set obtained by removing Jn \\ {xi} from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Likewise, define  Ai,j(x1, Jn) to be Ai,j(x1, Jn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21), but with each instance of Un−1(K) replaced by  ∑  µ⊢K′′  ∑  ⊔#µ  t=1Kt=Jn\\{xi,xj} ∏  µt∈µ  W#Kt+#µt(µt, Kt),  where K′′ is now the result of removing Jn \\ {xi, xj} from K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Finally, let Ai,j,h(x1, Jn) be the  function Ai,j,h(x1, Jn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) with each instance of Un−3(K) replaced by Wn−3(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Setting n = 2 in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) and subtracting equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) multiplied by U1(x2)  from the result yields the n = 2 loop equation on the connected correlators,  0 = x2  1  ∑  µ⊢(x1,x1,x1,x1)  ∑  ⊔#µ  t=1Kt={x2} ∏  µt∈µ  W#Kt+#µt(µt, Kt) − N3x1W2(x1, x2) +  ∂  ∂x2  A2(x1, x2)  + lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  � �  ∑  µ⊢(ξ,x1)  ∑  ⊔#µ  t=1Kt={x2} ∏  µt∈µ  W#Kt+#µt(µt, Kt)  + 1  2  ∑  µ⊢(x1,x1)  ∑  ⊔#µ  t=1Kt={x2} ∏  µt∈µ  W#Kt+#µt(µt, Kt)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29)  252  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Hermitised Laguerre Ensemble  the auxiliary function A2(x1, x2) is as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting the large N  expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) into this equation, multiplying the result by N−3, and taking the limit  N → ∞ gives the (n, l) = (2, 0) loop equation  0 = 4x2  1  �  W0  1(x1)  �3 W0  2(x1, x2) − x1W0  2(x1, x2) −  ∂  ∂x2  x2  2  �  W0  1(x2)  �3  x1 − x2  +  ∂  ∂x2  x1x2  �  3  �  W0  1(x1)  �3 −  �  W0  1(x1)  �2 W0  1(x2) − W0  1(x1)  �  W0  1(x2)  �2�  x1 − x2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30)  As seen in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, the above method for deriving equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) extends to the general  n ⩾ 3 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, let us now give the analogue of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 for the global scaled (N, N)  Hermitised Laguerre ensemble — we omit the proof, as it is exactly the same as that given  for Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23) now assuming the role of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let Ai(x1, Jn), Ai,j(x1, Jn), and Ai,j,h(x1, Jn) be as introduced in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For n ⩾ 3, the connected correlators of the global scaled (N, N) Hermitised Laguerre ensemble  specified by equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) satisfy the loop equation  0 = x2  1  ∑  µ⊢(x1,x1,x1,x1)  ∑  ⊔#µ  t=1Kt=Jn  ∏  µt∈µ  W#Kt+#µt(µt, Kt) − N3x1Wn(x1, Jn) +  n  ∑  i=2  ∂  ∂x2  Ai(x1, Jn)  +  ∑  2⩽i<j⩽n  ∂2  ∂xi∂xj  Ai,j(x1, Jn) + 6  x1  ∑  2⩽i<j<h⩽n  ∂3  ∂xi∂xj∂xh  Ai,j,h(x1, Jn)  + lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  � �  ∑  µ⊢(ξ,x1)  ∑  ⊔#µ  t=1Kt=Jn  ∏  µt∈µ  W#Kt+#µt(µt, Kt)  + 1  2  ∑  µ⊢(x1,x1)  ∑  ⊔#µ  t=1Kt=Jn  ∏  µt∈µ  W#Kt+#µt(µt, Kt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31)  The above loop equation can be used to derive loop equations characterising the correlator  expansion coefficients Wl  n using the method described below Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2,  we do not display these loop equations here, but note that they are straightforward rewritings  of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) in a similar fashion to how equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) is just equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) with  products of connected correlators Wn′ replaced by suitable sums of products of correlator  expansion coefficients Wl′  n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) above is similar to the analogous equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) for the  (N, N) antisymmetrised Laguerre ensemble in that, in the same sense as described below  253  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, it is of higher order than the loop equations for the classical and general  β ensembles reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Points of difference include the occurrence of the limiting  term in the last two lines of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) and the lack of denominators involving  differences of squares, which is a special feature of the antisymmetrised Laguerre ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moreover, the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) is in fact higher order than equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) (and also  the loop equation of [74]), this being indicated by the spectral curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) being a quartic  polynomial in W0  1(x1) instead of cubic, the first sum in the right-hand side of equation  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) containing a product of four correlators instead of at most three, and there being a  sum over 2 ⩽ i < j < h ⩽ n of third order partial derivatives ∂xi∂xj∂xh of the simple functions  Ai,j,h(x1, Jn) involving Wn−3 in said loop equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the last two lines of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Discussion on W(H1),0  1  and W(H1),0  2  In parallel with §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, we now find a rational parametrisation for the spectral curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  of the global scaled (N, N) Hermitised Laguerre ensemble, evaluate W0  1(x1), W0  2(x1, x2), and  W2  1(x1) in terms of this parametrisation, and then use residue calculus to compute some  coefficients of the genus expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63)  ˜c(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = N2−n  3  2 (k1+···+kn)+1−n  ∑  l=0  c(H1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  Nl  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32)  for particular values of n, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, our immediate goal is to find rational functions  x(z), y(z) such that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) holds true for all z upon replacing W0  1(x1) by y(z) and  x1 by x(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' By using the Newton polygon method [32], we know that the complex algebraic  curve 0 = x2y4 − xy + 1 is of genus zero and is thus the Riemann sphere CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, our  desired parametrisation must exist (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', [286, Thrm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='63];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' an algorithm guaranteed to  produce such a parametrisation is given in [286, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Writing u(z) = x(z)y(z)2 shows  that our spectral curve (written in the x(z), y(z) variables) is equivalent to  0 = u(z)2 − u(z)  y(z) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  As this is a quadratic equation in u(z) of the same form as equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29), we may take  inspiration from the parametrisation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) and let  1  y(z) = z + 1  z =⇒ y(z) =  z  z2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33)  254  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Hermitised Laguerre Ensemble  Then, u(z) = z (or u(z) = 1/z, but we ignore this choice without loss of generality) and,  consequently,  x(z) = u(z)  y(z)2 = (z2 + 1)2  z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  It can immediately be seen that the parametrisations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) satisfy the equation  0 = x(z)2y(z)4 − x(z)y(z) + 1,  so y(z) is the single-valued analytic continuation of W0  1(x(z)) to CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' That is, W0  1(x1) is equal  to y(z) when z ∈ x−1(x1) is chosen to be close to z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To see why we must make this choice,  note that our desired solution of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) has the property that W0  1(x1) ∼ 1/x1 as  x1 → ∞, which translates to the requirement that y(z) ∼ x(z)y(z)2 = u(z) = z → 0 as  x(z) → ∞ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the discussion below equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='51)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Moving on, taking z1, z2 ∈ CP1 and making the substitutions x1 = x(z1) and x2 = x(z2)  (with W0  1(xi) = y(zi)) into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='30) shows that  W0  2(x(z1), x(z2)) =  1  x′(z1)x′(z2)  1  (z1 − z2)2 −  1  (x(z1) − x(z2))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  This is exactly the expected universal form for W0  2 when working with a genus zero spectral  curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' see equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34) together with the discussion below it and cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the first line of  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Per the comment below equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28), W1  1(x1) is identically zero, so the next-to-leading  order term in the large N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) of W1(x1) is W2  1(x1), which is equivalently the  1/N2 correction to W0  1(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With the evaluation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35) in hand, we may set l = 2 in the  loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) with x1 replaced by x(z1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34), W0  1(x1) replaced by y(z1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33),  and W0  2(x1, x1) replaced by the limit  lim  z2→z1 W0  2(x(z1), x(z2)) = 6z6  1 − 2z4  1 + 10z2  1 + 2  z4  1 x′(z1)4  to compute  W2  1(x(z1)) = −z3  1(9z8  1 + 38z4  1 + 16z2  1 + 1)  (3z2  1 − 1)5(z2  1 + 1)3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36)  As in the case of the antisymmetrised Laguerre ensemble, comparing the genus expansion  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) of ˜c(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn to the large x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29) and large N expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) of  255  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  the connected correlators Wn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) = ˜W(H1)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) shows that  Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) =  ∞  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn=0  c(H1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  xk1+1  1  · · xkn+1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37)  Hence, in analogy with equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='56), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57) and recalling that x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn) → ∞  corresponds to z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , zn → 0 in the regime where equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='29), consequently (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='37), is  valid, we may use the residue formulae  c(H1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = (−1)n  Res  x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',xn=∞xk1  1 · · · xkn  n Wl  n(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) dx1 · · · dxn  = (−1)n  Res  z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',zn=0Wl  n(x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn))  n  ∏  i=1  x(zi)kix′(zi) dzi  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  to compute the coefficients of the genus expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) into the latter residue formula, we present the evaluation of c(H1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn for  (n, l) = (1, 0), (1, 2), (2, 0) and some low values of k1, k2 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let us mention  here the particular values c(H1),0  4  = 4, c(H1),2  2  = 1, and c(H1),0  1,1  = 2, which tell us that there  exist four distinct genus zero rooted topological hypermaps with one degree four black  vertex satisfying the criteria of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' there is one genus one orientable rooted  topological hypermap with a degree four black vertex satisfying said criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and there are  two genus zero 2-rooted topological hypermaps with two bivalent black vertices of the type  defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We display these topological hypermaps in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5: The four topological hypermaps in the top row are counted by c(H1),0  4  , the  toroidal hypermap in the bottom left corresponds to c(H1),2  2  = 1, and the remaining two  topological hypermaps are enumerated by c(H1),0  1,1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Colours are as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  256  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Concluding Remarks and Outlook  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4  Concluding Remarks and Outlook  In this thesis, we have added onto the well established theory of classical matrix ensembles  (some of which is reviewed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), elaborated on the relatively under-reported but  known combinatorial theory of said ensembles, and then constructed some ribbon graphs  and loop equations for the Hermitised and antisymmetrised Laguerre ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Recall from  Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 that our studies have been in part motivated by the desire to better understand  1-point recursions, loop equations, and the conjectured [61] relationship between these two  recursive schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is a broad and difficult problem that we have partially addressed by  broadening the range of examples of 1-point recursions and loop equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' While this is far  from bringing said problem to resolution, our examples provide a source of open questions  that should stimulate meaningful progress in this endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, as we come to the end  of this thesis, let us tie it together with some suggestions on directions for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Larger matrix products and topological recursion  Recall that one of our original motivations for undertaking the loop equation analysis of  the global scaled (N, N) antisymmetrised and Hermitised Laguerre ensembles presented in  Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 was to explore the feasibility of treating the general m antisymmetrised  and Hermitised matrix products Jm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) and Hm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We posit that  the general procedure outlined in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 should be able to produce loop  equations for these matrix product ensembles, but note that the outcomes are expected to be  prohibitively convoluted for large values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, the m > 1 extensions of equations  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) require us to consider the action of differential operators of the form  ∂(XTm)imim−1 · · · ∂(XT  1 )i1i0(JT)i0j0∂(X1)j0j1 · · · ∂(Xm)jm−1jm  with Xi, J defined analogously to the X, J of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and  ∂(X†m)imim−1 · · · ∂(X†  1)i1i0∂( ˜H†)i0j0∂(X1)j0j1 · · · ∂(Xm)jm−1jm  with Xi, ˜H defined analogously to the X, ˜H of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Consequently, we  anticipate that the analogue of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) pertaining to some fixed m > 1 should  contain a sum over partitions µ of the (2m + 1)-tuple (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x1) of the form seen in the first  257  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  line of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45) and sums of up to (2m)th order partial derivatives of the form seen  in the final two lines of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='45);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' the m > 1 analogue of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='31) should  likewise contain a sum over partitions µ of the (2m + 2)-tuple (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x1) in addition to  sums of up to (2m + 1)th order partial derivatives of the canonical extensions of Ai,j,h(x1, Jn)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22) having up to (2m + 1) indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It was mentioned in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 that applying residue calculus to the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) for  the correlator expansion coefficients Wl  n of the one-cut 1-Hermitian matrix model to obtain  the simpler topological recursion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='41) for the associated differential forms ωl  n  essentially removes the need for considering the terms Pl  n(x1, Jn) and  n  ∑  i=2  ∂  ∂xi  �  Wl  n−1(x1, Jn \\ {xi}) − Wl  n−1(Jn)  x1 − xi  �  seen on the right-hand side of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, it stands to reason that one  option for alleviating the aforementioned complexities associated with the loop equations  for the general m antisymmetrised and Hermitised matrix product ensembles is to use the  methodology reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 to instead derive topological recursion formulae for these  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This would very likely either replace the aforementioned general m analogues of  Ai,j,h(x1, Jn) with drastically simpler versions of themselves or possibly remove the need for  considering them altogether — it was commented in [74] that the loop equations therein for  the (N, N, N) complex Wishart product ensemble are also expected to reduce to relatively  simple topological recursion formulae upon applying the techniques of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It would  thus be very interesting to see if our loop equations (and those of [74]) can be simplified to  elegant topological recursion formulae in the proposed manner and, if so, to compare the  latter to the higher order topological recursion formulae of [45], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Although this would  still be far off from obtaining loop equations or topological recursion formulae for general m,  it should at the very least be possible to take N → ∞ and observe a uniform structure across  all m ∈ N, thereby enabling us to recover the spectral curves known from equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) [29], [273], [127], [143], [144],  0 = x2m−1  1  �  W(Jm),0  1  (x1)  �2m+1  + x1W(Jm),0  1  (x1) − 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  0 = x2m  1  �  W(Hm),0  1  (x1)  �2m+2  − x1W(Jm),0  1  (x1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  258  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Concluding Remarks and Outlook  As an aside, a related endeavour that may be worth pursuing is to see if the theory of  [86], [63] concerning spectral curves corresponding to eigenvalue densities with hard edges  and that of [108] relating to topological recursion for general β ensembles can be used to  derive explicit topological recursion formulae from the loop equations [142] for the Laguerre  and Jacobi β ensembles discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Such explicit formulae should be attainable as  these ensembles fall within the abstract setting considered in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  On 1-point recursions  As the loop equation formalism and topological recursion fully determine the resolvent  expansion coefficients Wl  1, which act as generating functions for the moment expansion  coefficients  ˜Mk,l defined implicitly through equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32), it is reasonable to say that  the data produced by 1-point recursions on the  ˜Mk,l is a subset of that produced by the  loop equation formalism — the benefit in considering 1-point recursions is that they are  more efficient at generating this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, it should come as no surprise that many  ensembles that can be studied using 1-point recursions are also known to be susceptible to  loop equation analysis (see [61] and references therein for a brief review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Seeing as how the  1-point recursions given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 characterise random matrix ensembles that have also been  shown to be governed by the loop equations reviewed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, it is natural to ask if there  exist 1-point recursions on the ˜Mk,l = cl  k that are determined by the loop equation analysis  of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We suggest two approaches for obtaining such 1-point recursions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It may be possible to derive 1-point recursions for the antisymmetrised Laguerre  ensemble by using the techniques of Chapters 2 and 3 in addition to the theory of [125]  relating the Laguerre Muttalib–Borodin ensemble to the Selberg integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' (Incidentally,  let us mention here that it remains to be seen how far the theory of Chapter 2 can be  pushed to uncover features of the classical β ensembles with general β ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It should be checked if Chekhov’s application [64] of an adaptation of the Br´ezin–  Hikami replica method [51] to derive the 1-point recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38) for the LUE with  a = 0 can be repurposed to treat the antisymmetrised and Hermitised Laguerre  ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It would likewise be interesting to see if the 1-point recursions of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 can  be obtained in this way, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  259  CHAPTER 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations for the Matrix Product Ensembles  Let us recall from the discussion at the end of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 that while the moment expansion  coefficients Mk,l of the Gaussian and Laguerre ensembles have combinatorial interpretations  due to the theory reviewed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2, these interpretations are not known to trans-  late to combinatorial interpretations for the 1-point recursions on said expansion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  However, preliminary experimentation by the present author suggests that dressing the  bijective proof of the Catalan recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25) sketched below Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 (and demonstrated  in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) with suitable additional rules may result in natural combinatorial proofs  of both the Harer–Zagier recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) and the 1-point recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24) for the LUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Confirming this conjecture to be true and comparing the proposed bijections to those men-  tioned in [61, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1] would shed light on how the 1-point recursions of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2 relate to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This is of particular interest to us since the 1-point recursions for the (shifted) JUE  presented in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 are very similar to those for the GUE and LUE — it is hoped that  a combinatorial understanding of the GUE and LUE 1-point recursions in terms of ribbon  graphs could extend to a similar understanding of the 1-point recursions and, in turn, the  spectral moments of the (shifted) JUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3  Combinatorics associated with the Jacobi unitary ensemble  In contrast to the approach described above for obtaining combinatorial interpretations for  the JUE moments through associated interpretations of its 1-point recursion, we now show  what happens if one tries to obtain such combinatorial interpretations by instead modifying  the arguments of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' First, let G1, G2 be drawn from the M1 × N, respectively  M2 × N, complex Ginibre ensemble of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 so that for i = 1, 2, Wi = G†  i Gi is drawn  from the (Mi, N) LUE, in accordance with Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, according to this same  definition, Y = W1(W1 + W2)−1 represents the (M1, M2, N) JUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Letting ⟨ · ⟩ denote averages  with respect to the product P(L)(W1)P(L)(W2), with P(L)(W) specified by equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2),  equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) then tells us that the spectral moments of the JUE are given by  m(JUE)  k  =  �  Tr Yk�  ,  k ∈ N  =  �  Tr  �  IN + W2W−1  1  �−k�  =  ∞  ∑  l=0  �l + k − 1  l  �  (−1)l �  Tr (W2W−1  1 )l�  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  260  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Concluding Remarks and Outlook  where we have put aside issues of convergence and used the generalised binomial theo-  rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This computation reduces the problem at hand to that of finding a combinatorial  interpretation for the spectral moments of W2W−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Using the independence of W1, W2, it can  be shown [326], [25] that said spectral moments are given by particular sums of products  of mixed moments of W2 and W−1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Alternatively, one may use the cyclic property of the  trace to show that these same spectral moments are given by the averages ⟨Tr (G2W−1  1 G†  2)l⟩,  which can be interpreted as the weighted count of LUE ribbon graphs, with the weights  being certain mixed moments of the inverse Wishart ensemble (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7 preceding  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The bottom line is that, since the mixed moments of the inverse Wishart  ensemble are currently not known to have any interpretations in terms of ribbon graphs, the  argument just given cannot produce such interpretations for the JUE spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  If one is content with combinatorial interpretations of JUE spectral moments that are  not related to ribbon graphs, then the situation is more fortuitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Indeed, the above  construction can be made fully combinatorial by observing that the mixed moments of the  inverse Wishart ensemble have recently been characterised in terms of monotone double  Hurwitz numbers [73], [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More recently still, the authors of [159] have characterised the  mixed moments of the JUE itself in terms of monotone triple Hurwitz numbers [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These  works complement the earlier work [40], which characterised the mixed moments of the  GUE in terms of so-called 2-orbifold strictly monotone Hurwitz numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A curious feature  of these works is that the proofs therein of the relationships between mixed moments and  Hurwitz numbers do not have a combinatorial flavour of the sort discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3,  even though Hurwitz numbers are very closely related to combinatorial (hyper)maps and  constellations (see [220], [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' This gap has been addressed in the GUE case [44], but it  remains to be seen how the Hurwitz number combinatorics of the LUE and JUE could be  related to ribbon graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It would be interesting to see if the results of [73], [159], [160]  can be reverse engineered to determine classes of ribbon graphs that are enumerated by  the mixed moments of the JUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' In that vein, the theory of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 could benefit from being  reformulated in terms of combinatorial hypermaps and/or constellations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' this would enable  the use of computer algebra for computing the relevant mixed cumulants and could also  reveal a connection between said cumulants and new classes of Hurwitz numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  261  Bibliography  [1] Digital Library of Mathematical Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' URL https://dlmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='gov/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 150  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Adachi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Toda, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Kubotani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Asymptotic analysis of singular values of rectangular  complex matrices in the Laguerre and fixed-trace ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A, 44(29):292002, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1088/1751-8113/44/29/292002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 62, 81, 94  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Adhikari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Reddy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Reddy, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Saha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Determinantal point processes in the  plane from products of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content=' In A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content=' Nonperturbative two-dimensional quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content=' Expo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
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+page_content='1007/s10955-015-1353-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 56  [328] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Zinn-Justin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Zuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' On some integrals over the U(N) unitary group and their large  N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A, 36(12):3173–3194, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1088/0305-4470/36/12/318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 58  [329] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Zvonkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Matrix integrals and map enumeration: An accessible introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Modelling, 26(8-10):281–304, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1016/S0895-7177(97)00210-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 63, 154  [330] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ˙Zyczkowski and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Sommers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Truncations of random unitary matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' A, 33(10):  2045–2057, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1088/0305-4470/33/10/307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 14  289  Appendices  Appendix A  Quaternionic Matrices and Pfaffians  The quaternions H are a number system that contain the real and complex numbers R, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  They are a finite-dimensional associative division algebra over R, of which there are exactly  three up to isomorphism (the other two being R, C) [146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' They are also referred to as a  skew-field since the only field axiom not satisfied by H is that of commutative multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The quaternions are best realised as the Clifford algebra Cl0,2(R), which is an R-algebra  generated by two elements that we label i and j, each squaring to negative one and together  satisfying the anti-commutation relation ij + ji = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Defining k := ij, one has the isomorphism  H ≃ {a + bi + cj + dk | a, b, c, d ∈ R}  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  with products induced by the following multiplication table:  ×  1  i  j  k  1  1  i  j  k  i  i  −1  k  −j  j  j  −k  −1  i  k  k  j  −i  −1  Note A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Some authors refer to the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) as the ‘real quaternions’ due to  the coefficients a, b, c, d being real, in contrast to the ‘complex quaternions’ defined by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  with a, b, c, d ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Throughout this thesis, H refers strictly to the real quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  291  APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Quaternionic Matrices and Pfaffians  We can realise the well known embedding MM×N(H) �→ M2M×2N(C) by mapping each  matrix entry according to  a + bi + cj + dk �→  �  � a + bi  c + di  −c + di  a − bi  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  In fact, we identify MM×N(H) with the set of M × N matrices whose entries are themselves  2 × 2 matrices of the form given on the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The quaternionic conjugate of q = a + bi + cj + dk is q = a − bi − cj − dk and the squared  norm is |q|2 = qq = qq = a2 + b2 + c2 + d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The quaternionic dual of a 2N × 2M complex  matrix X is given by  XD := −J2MXTJ2N  where  J2N := IN ⊗  �  �0  −1  1  0  �  �  is the 2N × 2N block-diagonal matrix with the diagonal blocks given by  J2 =  �  �0  −1  1  0  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  If Q is the matrix representation of the quaternion q, then QD is the matrix representation of  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' More generally, if X ∈ MM×N(H), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=', if X is a 2M × 2N complex matrix with 2 × 2 block  structure such that each block can be interpreted as a quaternion, then XD is the quaternionic  adjoint X† of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Here, the quaternionic adjoint X† is equivalent to the transpose of the  quaternionic conjugate of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' With this in mind, the (unitary) symplectic group is defined by  an obvious generalisation of the orthogonal and unitary groups,  Sp(2N) := {U ∈ MN×N(H) | U†U = I2N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  Due to the fact that H is not commutative, the determinant is not well-defined for  quaternionic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' When deciding on a definition, one needs to choose which properties  of the determinant must be retained and which can be jettisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For our purposes, it is  enough that the determinant correspond to the product of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we adopt a  definition of Dyson’s for self-adjoint quaternionic matrices [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  292  APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Quaternionic Matrices and Pfaffians  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Let X ∈ MN×N(H) be self-adjoint and for the sake of clarity, let ˜X be its  2N × 2N complex-matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, define the quaternionic determinant as  Det X := Pf J−1  2N ˜X,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  where Pf Y is the Pfaffian of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, to ensure that the trace of a quaternionic matrix is  the sum of its eigenvalues, we define the quaterionic trace as  Tr X :=  N  ∑  i=1  Re (Xii) = 1  2 Tr ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The Pfaffian of an anti-symmetric matrix can be thought of as the square root of its  determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Strictly speaking, given an anti-symmetric matrix X ∈ M2N×2N(C),  Pf X =  1  2NN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ∑  σ∈S2N  sgn(σ)Xσ(1),σ(2)Xσ(3),σ(4) · · · Xσ(2N−1),σ(2N),  where S2N is the symmetric group of order (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and sgn(σ) is the sign of the permutation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Similar to the determinant, the Pfaffian can alternatively be computed through the following  cofactor expansion (expanding along the first row):  Pf X =  2N  ∑  j=2  (−1)jX1j Pf X(1, j),  where the minor X(1, j) is obtained from X by deleting the first and jth rows and columns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' if  one were instead computing the determinant, only the first row and jth column of X would  need to be deleted to produce the desired minor X(1, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' The Pfaffian has an interpretation  as a square root because of the fact that (Pf X)2 = Det X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that this means that the  square of the right-hand side of equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) is Det ˜X, so that eigenvalues are not  double-counted when computing the quaternionic determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  293  Appendix B  Particular Stieltjes Transforms  To take the Stieltjes transforms of equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12), we need to compute terms  of the form  � 1  0  xp(1 − x)q  s − x  dn  dxn ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) dx  for β = 2 or 4, 0 ⩽ n ⩽ 5, n ⩽ q ⩽ n + 2, and q ⩽ p ⩽ q + 1 all integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' To this end, we  define  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, n, k) :=  � 1  0  xp(1 − x)q  (s − x)k  dn  dxn ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) dx  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  for integers 0 ⩽ n ⩽ q ⩽ p and k ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, integration by parts gives the identity  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, n, k) = (p + q)Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q − 1, n − 1, k) − pIβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p − 1, q − 1, n − 1, k)  − kIβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, n − 1, k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  Applying this identity n times allows us to reduce Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, n, k) to an expression involving  terms of the form Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, 0, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Then, considering (s − x)−k−1 = (−1)k/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' ∂k  s(s − x)−1 for  k ⩾ 0 gives us  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, 0, k + 1) = (−1)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  dk  dsk Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, 0, 1),  k ⩾ 0,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  which allows us to further reduce to an expression involving terms of the form Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Finally, factorisation of xp − sp for positive integer p yields  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, 0, 1) = spIβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 0, q, 0, 1) −  p−1  ∑  l=0  sp−l−1Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' l, q, 0, 0),  p ⩾ 1  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  294  APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Particular Stieltjes Transforms  and likewise  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 0, q, 0, 1) =  q  ∑  m=0  � q  m  �  (−1)m  �  smW(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) −  m−1  ∑  l=0  sm−l−1m(J)  l  �  ,  q ⩾ 1,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  where m(J)  l  is the lth spectral moment of ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' These last two equations thus reduce  our expression to one involving powers of s, derivatives of W(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β), and spectral  moments of ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To begin, we list  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 1, 1, 1, 1) = s(1 − s) d  dsW(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) − N  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 2, 2, 2, 1) = s2(1 − s)2 d2  ds2W(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) − 2N(s − 2) − 6m(J)  1  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 3, 3, 3, 1) = s3(1 − s)3 d3  ds3W(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) − 6N  �  s2 − 3s + 3  � − 24m(J)  1 (s − 3) − 60m(J)  2  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 4, 4, 4, 1) = s4(1 − s)4 d4  ds4W(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) − 24N(s3 − 4s2 + 6s − 4)  − 120m(J)  1  �  s2 − 4s + 6  � − 360m(J)  2 (s − 4) − 840m(J)  3  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' 5, 5, 5, 1) = s5(1 − s)5 d5  ds5W(J)  1 (s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) − 120N(s4 − 5s3 + 10s2 − 10s + 5)  − 720m(J)  1 (s3 − 5s2 + 10s − 10) − 2520m(J)  2 (s2 − 5s + 10)  − 6720m(J)  3 (s − 5) − 15120m(J)  4  All of the necessary Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' p, q, n, k) can be obtained from these through variants of identity  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) and integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' For example,  Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' n + 1, n, n, 1) = sIβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' n, n, n, 1) − Iβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' n, n, n, 0)  = sIβ(s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' n, n, n, 1) + (−1)n+1  � 1  0  dn  dxn (xn(1 − x)n)ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β) dx,  which only requires knowledge of the spectral moments m(J)  0  to m(J)  n  of ρ(J)(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' N, β), in  addition to a term from the above list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' It should be noted that applying the Stieltjes transform  to xp(1 − x)q dn  dxn ρ(J)(x) produces sp(1 − s)q dn  dsn W(J)  1 (s) plus terms that do not involve W(J)  1 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  It follows that the differential equations for the resolvents will be the same as those for the  eigenvalue densities, with additional inhomogeneous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  295  Appendix C  Loop Equations on Moments  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Loop Equations on the ˜m(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  Let mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = ˜m(J1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn be as defined in equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Substituting equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)–  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26) into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) and then substituting the result into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15) yields  the following loop equation (recall that In = (k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn):  0 = N2mk1+1,In +  ∑  p1+p2+p3=k1−1  mp1,p2,p3,In −  ∑  p1+p2=k1−1  mp1,p2,In + k2  1 − 1  2  mk1−1,In  + 8  ∑  2⩽i<j⩽n  kikj[ki + 1]mod 2[kj + 1]mod 2mk1+ki+kj−1,In\\{ki,kj}  + 2  n  ∑  i=2  ki[ki + 1]mod 2  �  2  ∑  p1+p2=k1−1  mki+p1,p2,In\\{ki} +  ∑  p1+p2=ki���1  mk1+p1,p2,In\\{ki}  − mk1+ki−1,In\\{ki}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  To convert this equation into the loop equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) on the unconnected correlators  Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) = ˜U(J1)  n  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), we multiply both sides of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) by  x−k1−1  1  · · x−kn−1  n  and sum over k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Applying this prescription term by term  requires the following glossary (recall that Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)):  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ∑  p1+p2+p3=k1−1  mp1,p2,p3,In  xk1+1  1  · · xkn+1  n  = x1Un+2(x1, x1, x1, Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ∑  p1+p2=k1−1  mp1,p2,In  xk1+1  1  · · xkn+1  n  = Un+1(x1, x1, Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  296  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  mk1+1,In  xk1+1  1  · · xkn+1  n  = x1Un(x1, Jn) − NUn−1(Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  k2  1 − 1  2  mk1−1,In  xk1+1  1  · · xkn+1  n  =  1  2x1  �  x2  1  ∂2  ∂x2  1  + x1  ∂  ∂x1  − 1  �  Un(x1, Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ki [ki + 1]mod 2  mk1+ki−1,In\\{ki}  xk1+1  1  · · xkn+1  n  = 1  x1  ∂  ∂xi  � x2  i Un−1(x1, Jn \\ {xi}) − x1xiUn−1(Jn)  x2  1 − x2  i  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ∑  p1+p2=ki−1  ki [ki + 1]mod 2  mk1+p1,p2,In\\{ki}  xk1+1  1  · · xkn+1  n  =  ∂  ∂xi  � x1xiUn(x1, Jn) − x2  i Un(xi, Jn)  x2  1 − x2  i  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ∑  p1+p2=k1−1  ki [ki + 1]mod 2  mki+p1,p2,In\\{ki}  xk1+1  1  · · xkn+1  n  =  ∂  ∂xi  � x2  i Un(x1, x1, Jn \\ {xi}) − x1xiUn(x1, Jn)  x2  1 − x2  i  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  kikj [ki + 1]mod 2  �  kj + 1  �  mod 2  mk1+ki+kj−1,In\\{ki,kj}  xk1+1  1  · · xkn+1  n  = 1  x1  ∂2  ∂xi∂xj  �  x1x2  i xjUn−2(Jn \\ {xi})  (x2  1 − x2  j )(x2  i − x2  j )  −  x1xix2  j Un−2(Jn \\ {xj})  (x2  1 − x2  i )(x2  i − x2  j )  +  x2  i x2  j Un−2(x1, Jn \\ {xi, xj})  (x2  1 − x2  i )(x2  1 − x2  j )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  The proofs of equations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)–(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) are a little more involved than one might expect,  so we compute a few illustrative examples for the reader’s convenience:  To prove equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), one needs to appropriately split up the factor xk1+1  1  and  interchange the order of summation to see that  ∑  k1⩾0  ∑  p1+p2+p3=k1−1  mp1,p2,p3,In  xk1+1  1  = x1  ∞  ∑  k1=0  k1−1  ∑  p1=0  k1−p1−1  ∑  p2=0  mp1,p2,k1−p1−p2−1,In  xp1+1  1  xp2+1  1  xk1−p1−p2  1  = x1  ∞  ∑  p1=0  ∞  ∑  k1=p1+1  k1−p1−1  ∑  p2=0  mp1,p2,k1−p1−p2−1,In  xp1+1  1  xp2+1  1  xk1−p1−p2  1  = x1  ∞  ∑  p1=0  ∞  ∑  k1=0  k1  ∑  p2=0  mp1,p2,k1−p2,In  xp1+1  1  xp2+1  1  xk1−p2+1  1  297  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  = x1  ∞  ∑  p1=0  ∞  ∑  p2=0  ∞  ∑  k1=p2  mp1,p2,k1−p2,In  xp1+1  1  xp2+1  1  xk1−p2+1  1  = x1  ∞  ∑  p1=0  ∞  ∑  p2=0  ∞  ∑  k1=0  mp1,p2,k1,In  xp1+1  1  xp2+1  1  xk1+1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Multiplying this result by x−k2−1  2  · · x−kn−1  n  and further summing over k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 then  gives the right-hand side of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) has a simple but comparatively unique proof:  ∞  ∑  k1=0  mk1+1,In  xk1+1  1  =  ∞  ∑  k1=−1  mk1+1,In  xk1+1  1  − m0,In = x1  ∞  ∑  k1=−1  mk1+1,In  xk1+2  1  − m0,In  = x1  ∞  ∑  k1=0  mk1,In  xk1+1  1  − m0,In = x1  ∞  ∑  k1=0  mk1,In  xk1+1  1  − NmIn,  using the fact that m0,In =  �  Tr B0 Tr BIn� = N  �  Tr BIn� = NmIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Multiplying again by  x−k2−1  2  · · x−kn−1  n  and summing over k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 produces the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  To prove equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), we first note that since [ki + 1]mod 2 vanishes if ki is odd, we  can replace ki by 2ki in the summand of the left-hand side of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' However,  since mk1+2ki−1,In\\{ki} =  �  Tr Bk1+2ki−1 Tr BIn\\{ki}�  vanishes whenever k1 is even, we must also  replace k1 by 2k1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, the left-hand side of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6) simplifies as  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ki [ki + 1]mod 2  mk1+ki−1,In\\{ki}  xk1+1  1  · · xkn+1  n  = − ∂  ∂xi  xi  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  m2k1+2ki,In\\{ki}  x2k1+2  1  xk2+1  2  · · xki−1+1  i−1  x2ki+1  i  xki+1+1  i+1  · · xkn+1  n  = − 1  x1  ∂  ∂xi  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  m2k1+2ki,In\\{ki}  x2k1+2ki+1  1  xk2+1  2  · · xki−1+1  i−1  xki+1+1  i+1  · · xkn+1  n  � x1  xi  �2ki  = − 1  x1  ∂  ∂xi  ∑  k2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0 ∑  k1⩾ki  m2k1,In\\{ki}  x2k1+1  1  xk2+1  2  · · xki−1+1  i−1  xki+1+1  i+1  · · xkn+1  n  � x1  xi  �2ki  = − 1  x1  ∂  ∂xi  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ki−1,ki+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  k1  ∑  ki=0  m2k1,In\\{ki}  x2k1+1  1  xk2+1  2  · · xki−1+1  i−1  xki+1+1  i+1  · · xkn+1  n  � x1  xi  �2ki  = − 1  x1  ∂  ∂xi  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ki−1,ki+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  m2k1,In\\{ki}  x2k1+1  1  xk2+1  2  · · xki−1+1  i−1  xki+1+1  i+1  · · xkn+1  n  1 − (x1/xi)2k1+2  1 − x2  1/x2  i  = 1  x1  ∂  ∂xi  x2  i  x2  1 − x2  i  �  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ki−1,ki+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  m2k1,In\\{ki}  x2k1+1  1  xk2+1  2  · · xki−1+1  i−1  xki+1+1  i+1  · · xkn+1  n  −  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',ki−1,ki+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  x1  xi  m2k1,In\\{ki}  x2k1+1  i  xk2+1  2  · · xki−1+1  i−1  xki+1+1  i+1  · · xkn+1  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  298  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  We may now replace 2k1 by k1 in the summands without any problem since mk1,In\\{ki} = 0  whenever k1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Rewriting the sums in terms of the unconnected correlators according  to equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and performing some basic algebra then yields the right-hand side of  equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  We do not display the proofs of equations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)–(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9) since they can  be proven using extensions of the ideas given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Loop Equations on the ˜m(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn  Let mk1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn and Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) now be as defined in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Taking i, j, h to be pairwise distinct, the averages Ai′,j′,h′ of equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) are given by  A1,1,1 =  ∑  p1+···+p4=k1−1  mp1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',p4,In + k1(k1 − 1)  2  ∑  p1+p2=k1−1  mp1,p2,In  + 2  ∑  p1+p2+p3=k1−1  p1mp1+p2,p3,In,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  A1,1,i = A1,i,1 = Ai,1,1  = kik1(k1 − 1)  2  mk1+ki−1,In\\{ki} + ki  ∑  p1+p2+p3=k1−1  mp1+ki,p2,p3,In\\{ki},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  A1,i,i = Ai,i,1 = ki  ∑  p1+p2=k1−1  ∑  q1+q2=ki−1  mp1+q1+1,p2,q2,In\\{ki},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  Ai,1,i = k1k2  i mk1+ki−1,In\\{ki},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  Ai,i,i = k2  i (ki − 1)  2  mk1+ki−1,In\\{ki} + ki  ∑  p1+p2+p3=ki−1  mk1+p1,p2,p3,In\\{ki},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  A1,i,j = Ai,j,1 = kikj  ∑  p1+p2=k1−1  mp1+ki+kj,p2,In\\{ki,kj},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  Ai,1,j = kikj  ∑  p1+p2=k1−1  mp1+ki,p2+kj,In\\{ki,kj},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='7)  Ai,i,j = Aj,i,i = kikj  ∑  p1+p2=ki−1  mk1+p1+kj,p2,In\\{ki,kj},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8)  Ai,j,i = kikj  ∑  p1+p2=ki−1  mk1+p1,p2+kj,In\\{ki,kj},  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='9)  Ai,j,h = kikjkhmk1+ki+kj+kh−1,In\\{ki,kj,kh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10)  299  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  We do not provide proofs for equations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)–(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10), as they can be computed using  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 via the same ideas as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Now, substituting equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19) into equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) while observing that some of  the above expressions are unchanged upon reordering indices shows that  0 = −N3mk1+1,In + k1  ∑  p1+p2=k1−1  mp1,p2,In + 2  ∑  2⩽i<j⩽n  kikjmk1,ki+kj−1,In\\{ki,kj}  +  n  ∑  i=2  ki  �  2k1mk1+ki−1,In\\{ki} +  ∑  p1+p2=ki−1  mk1,p1,p2,In\\{ki}  �  + A1,1,1  +  n  ∑  i=2  (3A1,1,i + 2A1,i,i + Ai,1,i + Ai,i,i) + 2  ∑  2⩽i<j⩽n  �  2A1,i,j + Ai,1,j  �  + ∑  2⩽i,j⩽n,  i̸=j  �  2Ai,i,j + Ai,j,i  � + 6  ∑  2⩽i<j<h⩽n  Ai,j,h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='11)  Inserting the expressions (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)–(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='10) into this equation then produces the loop equation  on the mixed moments,  0 = −N3mk1+1,In + k1  ∑  p1+p2=k1−1  mp1,p2,In + 2  ∑  2⩽i<j⩽n  kikjmk1,ki+kj−1,In\\{ki,kj}  +  n  ∑  i=2  ki  �  2k1mk1+ki−1,In\\{ki} +  ∑  p1+p2=ki−1  mk1,p1,p2,In\\{ki}  �  +  ∑  p1+···+p4=k1−1  mp1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  + k1(k1 − 1)  2  ∑  p1+p2=k1−1  mp1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In + 2  ∑  p1+p2+p3=k1−1  p1mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  + 3  n  ∑  i=2  �  kik1(k1 − 1)  2  mk1+ki−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki} + ki  ∑  p1+p2+p3=k1−1  mp1+ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki}  �  +  n  ∑  i=2  �  2ki  ∑  p1+p2=k1−1  ∑  q1+q2=ki−1  mp1+q1+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki} + k1k2  i mk1+ki−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki}  �  +  n  ∑  i=2  �  k2  i (ki − 1)  2  mk1+ki−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki} + ki  ∑  p1+p2+p3=ki−1  mk1+p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki}  �  + 2  ∑  2⩽i<j⩽n  kikj  �  2  ∑  p1+p2=k1−1  mp1+ki+kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='kj} +  ∑  p1+p2=k1−1  mp1+ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2+kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='kj}  �  + ∑  2⩽i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='j⩽n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  i̸=j  kikj  �  2  ∑  p1+p2=ki−1  mk1+p1+kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='kj} +  ∑  p1+p2=ki−1  mk1+p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2+kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='kj}  �  + 6  ∑  2⩽i<j<h⩽n  kikjkhmk1+ki+kj+kh−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In\\{ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='kj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='kh}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12)  300  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  Multiplying both sides of this equation by x−k1−1  1  · · x−kn−1  n  and then taking the sum over  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 yields the loop equation on the unconnected correlators Un(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn) given  in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We apply this prescription term by term using extensions of the ideas  underlying the proofs presented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 above (recall again that Jn = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , xn)):  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  mk1+1,In  xk1+1  1  · · xkn+1  n  = x1Un(x1, Jn) − NUn−1(Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  k1  ∑  p1+p2=k1−1  mp1,p2,In  xk1+1  1  · · xkn+1  n  = − ∂  ∂x1  x1Un+1(x1, x1, Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  kikj  mk1,ki+kj−1,In\\{ki,kj}  xk1+1  1  · · xkn+1  n  =  ∂2  ∂xi∂xj  � xiUn−1(x1, Jn \\ {xi}) − xjUn−1(x1, Jn \\ {xj})  xi − xj  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  k1ki  mk1+ki−1,In\\{ki}  xk1+1  1  · · xkn+1  n  =  ∂2  ∂x1∂xi  � x1Un−1(Jn) − xiUn−1(x1, Jn \\ {xi})  x1 − xi  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  ki  ∑  p1+p2=ki−1  mk1,p1,p2,In\\{ki}  xk1+1  1  · · xkn+1  n  = − ∂  ∂xi  xiUn+1(x1, xi, Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  A1,1,1  xk1+1  1  · · xkn+1  n  = x2  1Un+3(x1, x1, x1, x1, Jn)  +  �1  2x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  + 1  �  Un+1(x1, x1, Jn)  + lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  �  Un+1(ξ, x1, Jn),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  A1,1,i  xk1+1  1  · · xkn+1  n  =  ∂  ∂xi  x1xi  x1 − xi  �  Un+1(x1, x1, x1, Jn \\ {xi}) − Un+1(x1, x1, Jn)  �  + 1  2  ∂3  ∂x2  1∂xi  � x1xiUn−1(x1, Jn \\ {xi}) − x2  1Un−1(Jn)  x1 − xi  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  A1,i,i  xk1+1  1  · · xkn+1  n  =  ∂  ∂xi  � x1xiUn+1(x1, x1, Jn) − x2  i Un+1(x1, xi, Jn)  x1 − xi  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='20)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  Ai,1,i  xk1+1  1  · · xkn+1  n  =  �  xi  ∂  ∂xi  + 1  �  ∂2  ∂x1∂xi  � xiUn−1(x1, Jn \\ {xi}) − x1Un−1(Jn)  x1 − xi  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='21)  301  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  Ai,i,i  xk1+1  1  · · xkn+1  n  =  1  2x1  �  xi  ∂  ∂xi  + 1  � ∂2  ∂x2  i  � x2  i Un−1(x1, Jn \\ {xi}) − x1xiUn−1(Jn)  x1 − xi  �  + ∂  ∂xi  x2  i  x1 − xi  �  Un+1(x1, xi, Jn) − Un+1(xi, xi, Jn)  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  A1,i,j  xk1+1  1  · · xkn+1  n  =  ∂2  ∂xi∂xj  xixj  �Un−1(x1, Jn \\ {xi})  (x1 − xj)(xi − xj) − Un−1(x1, Jn \\ {xj})  (x1 − xi)(xi − xj)  + Un−1(x1, x1, Jn \\ {xi, xj})  (x1 − xi)(x1 − xj)  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='23)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  Ai,1,j  xk1+1  1  · · xkn+1  n  =  ∂2  ∂xi∂xj  xixj  (x1 − xi)(x1 − xj)  ×  �  Un−1(x1, x1, Jn \\ {xi, xj}) − Un−1(x1, Jn \\ {xj})  + Un−1(Jn) − Un−1(x1, Jn \\ {xi})  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='24)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  Ai,i,j  xk1+1  1  · · xkn+1  n  =  ∂2  ∂xi∂xj  xixj  �  Un−1(Jn)  (x1 − xj)(xi − xj) − Un−1(xi, Jn \\ {xj})  (x1 − xi)(xi − xj)  + Un−1(x1, Jn \\ {xj})  (x1 − xi)(x1 − xj)  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='25)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  Ai,j,i  xk1+1  1  · · xkn+1  n  =  ∂2  ∂xi∂xj  xixj  (x1 − xi)(xi − xj)  ×  �  Un−1(x1, Jn \\ {xj}) − Un−1(xi, Jn \\ {xj})  + Un−1(Jn) − Un−1(x1, Jn \\ {xi})  �  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='26)  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  Ai,j,h  xk1+1  1  · · xkn+1  n  = 1  x1  ∂3  ∂xi∂xj∂xh  � xixjxhUn−3(x1, Jn \\ {xi, xj, xh})  (x1 − xi)(x1 − xj)(x1 − xh)  − x1xjxhUn−3(Jn \\ {xj, xh})  (x1 − xi)(xi − xj)(xi − xh)  + x1xixhUn−3(Jn \\ {xi, xh})  (x1 − xj)(xi − xj)(xj − xh) −  x1xixjUn−3(Jn \\ {xi, xj})  (x1 − xh)(xi − xh)(xj − xh)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27)  Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) is simply equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4), while equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='14) can be obtained by  applying the operator − ∂  ∂x1 x1 to both sides of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Likewise, equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17)  can be obtained by first interchanging x1 ↔ xi in both sides of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) before  302  APPENDIX C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Loop Equations on Moments  applying − ∂  ∂xi xi to both sides of said equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16) is proven in a similar  way to equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6), with steps relating to the factor [ki + 1]mod 2 being skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' All  other computations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='15)–(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27), bar a caveat on equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18), can be proven using  the methods demonstrated in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1 above and the reasonings just given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' We now  conclude this appendix with a partial proof of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18):  The last term on the right-hand side of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) stands out in that it cannot be  treated in the same manner as the other expressions considered in this appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Multiplying  this term by x−k1−1  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' summing over k1 ⩾ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' and simplifying appropriately shows that  ∑  k1⩾0  2  ∑  p1+p2+p3=k1−1  p1  mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+1  1  = 2 ∑  k1⩾0  k1−1  ∑  p1=0  k1−p1−1  ∑  p2=0  p1  mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1−p1−p2−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+1  1  = 2  ∞  ∑  p1=0  ∞  ∑  k1=p1+1  k1−p1−1  ∑  p2=0  p1  mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1−p1−p2−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+1  1  = 2  ∞  ∑  p1=0  ∞  ∑  k1=0  k1  ∑  p2=0  p1  mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1−p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+p1+2  1  = 2  ∞  ∑  p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2=0  ∞  ∑  k1=p2  p1  mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1−p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+p1+2  1  = 2  ∞  ∑  p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1=0  p1  mp1+p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+p1+p2+2  1  = 2  ∞  ∑  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1=0  ∑  p1+p2=q  p1  mq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+1  1  xq+1  1  =  ∞  ∑  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1=0  q(q + 1) mq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='In  xk1+1  1  xq+1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Further multiplying this result by x−k2−1  2  · · x−kn−1  n  and summing over k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , kn ⩾ 0 then  shows that  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  2  ∑  p1+p2+p3=k1−1  p1  mp1+p2,p3,In  xk1+1  1  · · xkn+1  n  =  ∑  q,k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  q(q + 1)  mq,k1,In  xk1+1  1  xq+1  1  xk2+1  2  · · xkn+1  n  = lim  ξ→x1  �  x2  1  ∂2  ∂x2  1  + 2x1  ∂  ∂x1  �  ∑  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn⩾0  mq,k1,In  ξk1+1xq+1  1  xk2+1  2  · · xkn+1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  The final expression here is manifestly the third line of equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  303  Appendix D  Evaluation of Some Cumulants at  Leading Order  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1  Evaluation of c(J1),0  k1  , c(J1),1  k1  , and c(J1),0  k1,k2  As discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3, setting x(zi) = 1/(z3  i + zi) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='50) in the residue formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='57)  c(J1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = (−1)n  Res  z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',zn=0W(J1),l  n  (x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn))  n  ∏  i=1  x(zi)kix′(zi) dzi  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  and substituting in W(J1),0  1  (x(z1)) = z1 along with the specifications (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='52)  of, respectively, W(J1),1  1  (x(z1)) and W(J1),0  2  (x(z1), x(z2)) enables the computation of the  cumulant genus expansion coefficients c(J1),0  k1  , c(J1),1  k1  , and c(J1),0  k1,k2  defined implicitly through  equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Letting ˜J1 be drawn from the global scaled (N, N) antisymmetrised Laguerre ensemble  defined in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19, we have from equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='28) that  c(J1),0  k1  = lim  N→∞  1  N  �  Tr  ˜J k1  1  �  ,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  c(J1),1  k1  = lim  N→∞  ��  Tr  ˜J k1  1  �  − Nc(J1),0  k1  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  the aforementioned propositions also give combinatorial interpretations of these cumulant  expansion coefficients as signed enumerations of particular ribbon graphs and topological  hypermaps (see also Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Due to the antisymmetric nature of  ˜J1, the cumulant  expansion coefficients c(J1),0  k1  and c(J1),1  k1  vanish for odd values of k1, while for k1 even, the  304  APPENDIX D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Evaluation of Some Cumulants at Leading Order  former relates to the m = 2 Fuss–Catalan numbers (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) through the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='47)  c(J1),0  k1  = ˜M(J1)  k1,0 = ik1m(FC2)  k1/2 =  ik1  k1 + 1  �3k1/2  k1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  We now present the values of c(J1),0  k1  and c(J1),1  k1  computed through the residue formula (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  with 0 ⩽ k1 ⩽ 18 even — our calculations are consistent with equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  k1  0  2  4  6  8  10  12  14  16  18  c(J1),0  k1  1  −1  3  −12  55  −273  1 428  −7 752  43 263  −246 675  c(J1),1  k1  0  1  −5  28  −165  1 001  −6 188  38 760  −245 157  1 562 275  Comparing the second row of the above table to equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4) suggests that for k1 ∈ 2N,  c(J1),1  k1  = −ik1  �3k1/2 − 1  k1/2 − 1  �  + χk1=0 = χk1=0  3  − (k1 + 1)  3  c(J1),0  k1  ⇐⇒ W(J1),1  1  (x(z1)) =  x(z1)  3x′(z1)  ∂  ∂z1  W(J1),0  1  (x(z1)) +  1  3x(z1) =  x(z1)  3x′(z1) +  1  3x(z1),  which is precisely in keeping with equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For n = 2, the analogue of equations (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3) is  c(J1),0  k1,k2  = lim  N→∞  ��  Tr  ˜J k1  1 Tr  ˜J k2  1  �  −  �  Tr  ˜J k1  1  � �  Tr  ˜J k2  1  ��  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  see Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='19 in addition to Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4 for combinatorial interpretations  of this cumulant expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Note that the right-hand side of equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  is identically zero whenever either of k1, k2 are odd, since  ˜J1 is antisymmetric, and also  whenever either of k1, k2 equal zero, since Tr  ˜J 0  1 = N factors out of the first covariance  therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we present the values of c(J1),0  k1,k2  computed through equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) with  2 ⩽ k1, k2 ⩽ 14 even — the asterisks represent redundant data, as c(J1),0  k2,k1  = c(J1),0  k1,k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  k1  2  4  6  8  10  12  14  k2  c(J1),0  k1,k2  2  12  −80  504  −3 168  20 020  −127 296  813 960  4  ∗  600  −4 032  26 400  −171 600  1 113 840  −7 235 200  6  ∗  ∗  28 224  −190 080  1 261 260  −8 316 672  54 698 112  8  ∗  ∗  ∗  1 306 800  −8 808 800  58 810 752  −390 700 800  10  ∗  ∗  ∗  ∗  60 120 060  −405 437 760  2 715 913 200  12  ∗  ∗  ∗  ∗  ∗  2 756 976 768  −18 597 358 080  14  ∗  ∗  ∗  ∗  ∗  ∗  126 196 358 400  305  APPENDIX D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Evaluation of Some Cumulants at Leading Order  Comparing this table to Table 1 of [74] suggests that c(J1),0  k1,k2  relates to the analogous quantity  c(cW2),0  k1,k2  of the (N, N, N) complex Wishart product ensemble (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='12) according to  c(J1),0  2k1,2k2 = 4 (−1)k1+k2 c(cW2),0  k1,k2  ,  k1, k2 ∈ N,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  which is in agreement with Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2  Evaluation of c(H1),0  k1  , c(H1),2  k1  , and c(H1),0  k1,k2  In parallel to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1, let us recall from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3 that setting x(zi) = (z2  i + 1)2/zi (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='34)  in the residue formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='38)  c(H1),l  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn = (−1)n  Res  z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',zn=0W(H1),l  n  (x(z1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' , x(zn))  n  ∏  i=1  x(zi)kix′(zi) dzi  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1)  and then substituting in W(H1),0  1  (x(z1)) = y(z1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='33) along with the expressions for  W(H1),2  1  (x(z1)) and W(H1),0  2  (x(z1), x(z2)) given respectively in equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='36) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='35)  gives us a means to compute the coefficients c(H1),0  k1  , c(H1),2  k1  , and c(H1),0  k1,k2  of the genus expansions  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) of the associated mixed cumulants ˜c(H1)  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=',kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  With ˜H1 drawn from the global scaled (N, N) Hermitised Laguerre ensemble defined in  Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18, the relevant analogues of equations (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5) are  c(H1),0  k1  = lim  N→∞  1  N  �  Tr ˜Hk1  1  �  ,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)  c(H1),2  k1  = lim  N→∞  �  N  �  Tr ˜Hk1  1  �  − N2c(H1),0  k1  �  ,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3)  c(H1),0  k1,k2  = lim  N→∞  ��  Tr ˜Hk1  1 Tr ˜Hk2  1  �  −  �  Tr ˜Hk1  1  � �  Tr ˜Hk2  1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  Here, we have used the fact (recall the discussion following the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16)  that the l = 1 coefficient c(H1),1  k1  within the expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='32) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Like in the case of  the antisymmetrised Laguerre ensemble, the cumulant expansion coefficients (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2)–(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='4)  have combinatorial interpretations as enumerations of certain ribbon graphs and topological  hypermaps due to Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18 (see also Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  Recall from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5 that c(H1)  k1  , consequently ˜c(H1)  k1  and, in turn, c(H1),0  k1  and c(H1),2  k1  ,  vanish whenever k1 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Hence, we display the values of these expansion coefficients  obtained through equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1) for 0 ⩽ k1 ⩽ 18 even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  306  APPENDIX D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Evaluation of Some Cumulants at Leading Order  k1  0  2  4  6  8  10  12  14  16  18  c(H1),0  k1  1  1  4  22  140  969  7 084  53 820  420 732  3 362 260  c(H1),2  k1  0  1  34  645  9 828  133 620  1 694 154  20 490 470  239 545 800  2 729 482 668  In keeping with the discussion below equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='27), it is expected from the relation  W(H1),0  1  (x1) = x1W(FC3)  1  (x2  1)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5)  between the resolvents (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='18) of the limiting eigenvalue density ρ(H1),0(λ) of the global  scaled (N, N) Hermitised Laguerre ensemble (recall Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='13) and the m = 3 Fuss–  Catalan distribution (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='16), which is a consequence of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='22), that for even  integers k1, c(H1),0  k1  relates to the m = 3 Fuss–Catalan numbers (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='17) according to  c(H1),0  k1  = m(FC3)  k1/2 =  1  3k1/2 + 1  � 2k1  k1/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='6)  It can readily be checked that this is in line with the values of c(H1),0  k1  given in the first row of  the above table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  For the same reason as given below equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5), c(H1),0  k1,k2  vanishes when either of k1, k2  are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' Thus, we give evaluations of this cumulant expansion coefficient for 1 ⩽ k1, k2 ⩽ 9  — the asterisks have the same meaning as in the table for c(J1),0  k1,k2  displayed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  k1  1  2  3  4  5  6  7  8  9  k2  c(H1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k2  1  2  0  15  0  120  0  1 001  0  8 568  2  ∗  12  0  112  0  990  0  8 736  0  3  ∗  ∗  150  0  1 350  0  12 012  0  107 100  4  ∗  ∗  ∗  1 176  0  11 088  0  101 920  0  5  ∗  ∗  ∗  ∗  12 960  0  120 120  0  1 101 600  6  ∗  ∗  ∗  ∗  ∗  108 900  0  1 029 600  0  7  ∗  ∗  ∗  ∗  ∗  ∗  1 145 144  0  10 720 710  8  ∗  ∗  ∗  ∗  ∗  ∗  ∗  9 937 200  0  9  ∗  ∗  ∗  ∗  ∗  ∗  ∗  ∗  101 959 200  Observe that c(H1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='0  k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='k2  = 0 whenever k1 + k2 is odd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content=' which agrees precisely with what is  known from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}
+page_content='  307' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFJT4oBgHgl3EQfCCz-/content/2301.11428v1.pdf'}