SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 26 Giacomo Livan Marcel Novaes Pierpaolo Vivo Introduction to Random Matrices Theory and Practice 123 SpringerBriefs in Mathematical Physics Volume 26 Series editors Nathanaël Berestycki, Cambridge, UK Mihalis Dafermos, Princeton, USA Tohru Eguchi, Tokyo, Japan Atsuo Kuniba, Tokyo, Japan Matilde Marcolli, Pasadena, USA Bruno Nachtergaele, Davis, USA SpringerBriefs are characterized in general by their size (50–125 pages) and fast production time (2–3 months compared to 6 months for a monograph). Briefsareavailableinprintbutareintendedasaprimarilyelectronicpublicationto be included in Springer’s e-book package. Typical works might include: (cid:129) An extended survey of a field (cid:129) A link between new research papers published in journal articles (cid:129) Apresentation ofcoreconceptsthatdoctoral students mustunderstand inorder to make independent contributions (cid:129) Lecture notes making a specialist topic accessible for non-specialist readers. SpringerBriefsinMathematicalPhysicsshowcase,inacompactformat,topicsof current relevance in the field of mathematical physics. Published titles will encompassallareasoftheoreticalandmathematicalphysics.Thisseriesisintended for mathematicians, physicists, and other scientists, as well as doctoral students in related areas. More information about this series at http://www.springer.com/series/11953 Giacomo Livan Marcel Novaes (cid:129) Pierpaolo Vivo Introduction to Random Matrices Theory and Practice 123 Giacomo Livan Pierpaolo Vivo Department ofComputer Science Department ofMathematics University CollegeLondon King’sCollege London London London UK UK Marcel Novaes Instituto deFísica Universidade FederaldeUberlândia Uberlândia,Minas Gerais Brazil ISSN 2197-1757 ISSN 2197-1765 (electronic) SpringerBriefs inMathematical Physics ISBN978-3-319-70883-6 ISBN978-3-319-70885-0 (eBook) https://doi.org/10.1007/978-3-319-70885-0 LibraryofCongressControlNumber:2017958623 ©TheAuthor(s)2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This is a book for absolute beginners. If you have heard about random matrix theory,commonlydenotedRMT,butyoudonotknowwhatthatis,thenwelcome! this is the place for you. Our aim is to provide a truly accessible introductory account of RMT for physicists and mathematicians at the beginning of their researchcareer.Wetriedtowritethesortoftextwewouldhavelovedtoreadwhen we were beginning Ph.D. students ourselves. Our book is structured with light and short chapters, and the style is informal. The calculations we found most instructive are spelled out in full. Particular attentionispaid tothe numerical verification ofmost analytical results. The reader will find the symbol [♠test.m] next to every calculation/procedure for which a numerical verification is provided in the associated file test.m located at https:// github.com/RMT-TheoryAndPractice/RMT. We strongly believe that theory without practice is of very little use: In this respect, our book differs from most available textbooks on this subject (not so many, after all). Almost every chapter contains question boxes, where we try to anticipate and minimizepossiblepointsofconfusion.Also,weincludeToknowmoresectionsat the end of most chapters, where we collect curiosities, material for extra readings and little gems—carefully (and arbitrarily!) cherry-picked from the gigantic liter- ature on RMT out there. Ourbookcoversstandardmaterial—classicalensembles,orthogonalpolynomial techniques, spectral densities and spacings—but also more advanced and modern topics—replica approach and free probability—that are not normally included in elementary accounts on RMT. Due to space limitations, we have deliberately left out ensembles with complex eigenvaluesandmanyotherinterestingtopics.Ourbookisnotencyclopedic,noris itmeantasasurrogateorasummaryofotherexcellentexistingbooks.Whatweare sure about is that any seriously interested reader, who is willing to dedicate some oftheirtimetoreadandunderstandthisbooktilltheend,willnextbeabletoread and understand any other source (articles, books, reviews, tutorials) on RMT, without feeling overwhelmed or put off by incomprehensible jargon and endless series of “It can be trivially shown that…”. v vi Preface So,whatisarandommatrix?Well,itisjustamatrixwhoseelementsarerandom variables. No big deal. So why all the fuss about it? Because they are extremely useful! Just think in how many ways random variables are useful: If someone throws a thousand (fair) coins, you can make a rather confident prediction that the numberoftailswillnotbetoofarfrom500.Ok,maybethisisnotreallythatuseful, but it shows that sometimes it is far more efficient to forego detailed analysis of individual situations and turn to statistical descriptions. This is what statistical mechanics does, after all: It abandons the deterministic (predictive)lawsofmechanicsandreplacesthemwithaprobabilitydistributionon the space of possible microscopic states of your systems, from which detailed statistical predictions at large scales can be made. ThisiswhatRMT isabout,butinstead ofreplacing deterministic numberswith randomnumbers,itreplacesdeterministicmatriceswithrandommatrices.Anytime youneedamatrixwhichistoocomplicatedtostudy,youcantryreplacingitwitha random matrix and calculate averages (and other statistical properties). Anumberofpossibleapplicationscomeimmediatelytomind.Forexample,the Hamiltonian of a quantum system, such as a heavy nucleus, is a (complicated) matrix.ThiswasindeedoneofthefirstapplicationsofRMT,developedbyWigner. Rotationsarematrices;themetricofamanifoldisamatrix;theS-matrixdescribing the scattering of waves is a matrix; financial data can be arranged in matrices; matrices are everywhere. In fact, there are many other applications, some rather surprising, which do not come immediately to mind but which have proved very fruitful. WedonotprovideadetailedhistoricalaccountofhowRMTdeveloped,nordo we dwell too much on specific applications. The emphasis is on concepts, com- putations,tricksofthetrade:allyouneededtoknow(butwereafraidtoask)tostart a hopefully long and satisfactory career as a researcher in this field. Itisapleasuretothankhereallthepeoplewhohavesomehowcontributedtoour knowledge of RMT. We would like to mention in particular Gernot Akemann, Giulio Biroli, Eugene Bogomolny, Zdzisław Burda, Giovanni Cicuta, Fabio D. Cunden,PaoloFacchi,DavideFacoetti,GiuseppeFlorio,YanV.Fyodorov,Olivier Giraud, Claude Godreche, Eytan Katzav, Jon Keating, Reimer Kühn, Satya N. Majumdar, Anna Maltsev, Ricardo Marino, Francesco Mezzadri, Maciej Nowak, Yasser Roudi, Dmitry Savin, Antonello Scardicchio, Gregory Schehr, Nick Simm, Peter Sollich, Christophe Texier, Pierfrancesco Urbani, Dario Villamaina, and many others. This book is dedicated to the fond memory of Oriol Bohigas. London, UK Giacomo Livan Uberlândia, Brazil Marcel Novaes London, UK Pierpaolo Vivo Contents 1 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 One-Pager on Random Variables. . . . . . . . . . . . . . . . . . . . . . . 3 2 Value the Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Appetizer: Wigner’s Surmise. . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Eigenvalues as Correlated Random Variables . . . . . . . . . . . . . . 9 2.3 Compare with the Spacings Between i.i.d.’s. . . . . . . . . . . . . . . 9 2.4 Jpdf of Eigenvalues of Gaussian Matrices . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Classified Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 Count on Dirac. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Layman’s Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 To Know More... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 The Fluid Semicircle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.1 Coulomb Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Do It Yourself (Before Lunch) . . . . . . . . . . . . . . . . . . . . . . . . 25 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 Saddle-Point-of-View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 Saddle-Point. What’s the Point?. . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Disintegrate the Integral Equation . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Better Weak Than Nothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Smart Tricks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.5 The Final Touch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.6 Epilogue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.7 To Know More... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 vii viii Contents 6 Time for a Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.1 Intermezzo: A Simpler Change of Variables. . . . . . . . . . . . . . . 45 6.2 ...that Is the Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.3 Keep Your Volume Under Control . . . . . . . . . . . . . . . . . . . . . 46 6.4 For Doubting Thomases... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.5 Jpdf of Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . 48 6.6 Leave the Eigenvalues Alone. . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.7 For Invariant Models... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.8 The Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7 Meet Vandermonde. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.1 The Vandermonde Determinant . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2 Do It Yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 8 Resolve(nt) the Semicircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.1 A Bit of Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.2 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 8.3 Do It Yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.4 Localize the Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.5 To Know More... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 9 One Pager on Eigenvectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 10 Finite N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10.1 b¼2 is Easier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10.2 Integrating Inwards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 10.3 Do It Yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 10.4 Recovering the Semicircle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 11 Meet Andréief. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 11.1 Some Integrals Involving Determinants . . . . . . . . . . . . . . . . . . 75 11.2 Do It Yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 11.3 To Know More... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 12 Finite N Is Not Finished . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 12.1 b¼1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 12.2 b¼4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 13 Classical Ensembles: Wishart-Laguerre . . . . . . . . . . . . . . . . . . . . . 89 13.1 Wishart-Laguerre Ensemble. . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Contents ix 13.2 Jpdf of Entries: Matrix Deltas.... . . . . . . . . . . . . . . . . . . . . . . . 91 13.3 ...and Matrix Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 13.4 To Know More... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 14 Meet Marčenko and Pastur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 14.1 The Marčenko-Pastur Density . . . . . . . . . . . . . . . . . . . . . . . . . 97 14.2 Do It Yourself: The Resolvent Method . . . . . . . . . . . . . . . . . . 98 14.3 Correlations in the Real World and a Quick Example: Financial Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 15 Replicas... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 15.1 Meet Edwards and Jones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 15.2 The Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 15.3 Averaging the Logarithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 15.4 Quenched versus Annealed . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 16 Replicas for GOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 16.1 Wigner’s Semicircle for GOE: Annealed Calculation . . . . . . . . 109 16.2 Wigner’s Semicircle: Quenched Calculation . . . . . . . . . . . . . . . 112 16.2.1 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 16.2.2 One Step Back: Summarize and Continue . . . . . . . . . . 116 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 17 Born to Be Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 17.1 Things About Probability You Probably Already Know . . . . . . 119 17.2 Freeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 17.3 Free Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 17.4 Do It Yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124