Asymptotic Combinatorics with Applications to Mathematical Physics: A European Mathematical Summer School held at the Euler Institute, St. Petersburg, Russia July 9–20, 2001 PDF
Preview Asymptotic Combinatorics with Applications to Mathematical Physics: A European Mathematical Summer School held at the Euler Institute, St. Petersburg, Russia July 9–20, 2001
1815 Lecture Notes in Mathematics Editors: J.--M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris Subseries: EuropeanMathematicalSociety 3 Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo Anatoly M. Vershik (Ed.) Asymptotic Combinatorics with Applications to Mathematical Physics A European Mathematical Summer School held at the Euler Institute, St. Petersburg, Russia 9-20 2001 July , 1 3 Editor AnatolyM.Vershik St.PetersburgDepartment oftheMathematicalInstitute RussianAcademyofSciences Fontanka27 St.Petersburg191011,Russia e-mail:[email protected] http://www.pdmi.ras.ru/˜vershik TechnicalEditor YuriYakubovich St.PetersburgDepartment oftheMathematicalInstitute RussianAcademyofSciences Fontanka27 St.Petersburg191011,Russia e-mail:[email protected] Cataloging-in-PublicationDataappliedfor BibliographicinformationpublishedbyDieDeutscheBibliothek DieDeutscheBibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataisavailableintheInternetathttp://dnb.ddb.de MathematicsSubjectClassification(2000):46-XX,05-XX,60-XX,35-XX ISSN0075-8434 ISBN3-540-40312-4Springer-VerlagBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare liableforprosecutionundertheGermanCopyrightLaw. Springer-VerlagBerlinHeidelbergNewYorkamemberofBertelsmannSpringer Science+BusinessMediaGmbH http://www.springer.de (cid:1)c Springer-VerlagBerlinHeidelberg2003 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camera-readyTEXoutputbytheauthors SPIN:10933565 41/3142/du-543210-Printedonacid-freepaper Preface A European Mathematical Summer School entitled “ASYMPTOTIC COM- BINATORICS WITH APPLICATIONS TO MATHEMATICAL PHYSICS” was held at St. Petersburg, Russia, 9–22 July 2001. This meeting was at the same time a NATO Advanced Studies Institute. ItwascosponsoredbyNATOScienceCommittee,EuropeanMathematical Society, and Russian Fund for Basic Research. Thisvolumecontainsmathematicallecturesfromtheschool.Anotherpart ofthematerialspresentedattheSchool,morerelatedtomathematicalphysics, isalreadypublished1.InformationabouttheSchool,itsparticipants,program, etc. can be found at the end of this volume. The present volume contains lecture courses, as well as several separate lectures, which have mainly mathematical rather than physical orientation. They are aimed mostly at non-specialists and beginners who constituted the majority of the participants of the School. I would like to emphasize that splitting the lectures into “physical” and “mathematical” ones is relative. Moreover, the idea of the School was to unite mathematicians and physicists working essentially on the same problems but following different traditions and notations accepted by their communities. The last few years were marked by an impressive unification of a num- ber of areas in mathematical physics and mathematics. The Summer School presented some of these major – and until recently mutually unrelated – top- ics: matrix problems (the study of which was initiated by physicists about 25 years ago), asymptotic representation theory of classical groups (which arose in mathematics approximately at the same time), the theory of random matrices (also initiated by physicists but intensively studied by mathemati- cians), and, finally, the theory of integrable nonlinear problems in mathemat- ical physics with a wide range of related problems. As a result of the new 1 ProceedingsofNATOASIAsymptoticCombinatoricswithApplicationtoMathe- maticalPhysics,V.MalyshevandA.Vershik,Eds.,KluwerAcademicPublishers, 2002, 328pp. vi Preface interrelations dicovered, all these young theories, which constitute the essen- tialpartofmodernmathematicsandmathematicalphysics,havebecomepart of one large mathematical area. These interconnections are mainly combina- torial. An illustration of this phenomenon is the perception of the fact that theasymptotictheoryofYoungtableauxandthetheoryofspectraofrandom matrices is essentially the same theory, since the asymptotic microstructure ofarandomYoungdiagramwithrespecttothePlancherelmeasurecoincides with the microstructure of the spectrum of a random matrix in the Gaus- sian ensemble. The corresponding asymptotic distributions are new for the probability theory. They were originally found by Riemann–Hilbert problem techniques(theRiemann–Hilbertproblemarisesincalculationofthediagonal asymptotics of orthogonal polynomials). In the present volume this direction is represented by the lectures by P. Deift and A. Borodin. Later these dis- tributions were obtained by another method, that calculates the correlation functionsdirectlyusingdirectrelationstointegrableproblemsandhierarchies (A. Borodin, A. Okounkov, G. Olshansky). The lectures by A. Vershik, G. Olshansky, R. Hora, and partially by A.BorodinandP.Bianearedevotedtotheasymptoticrepresentationtheory. This theory studies the asymptotic behavior of characters of classical groups as the rank of the group grows to infinity. It was started in the beginning of 1970s by works of Vershik–Kerov and Logan–Shepp and one of the first results was the proof of the asymptotic behaviour of the characters of sym- metric group and Young diagrams. At that time the similarity and relations to quantum chromodynamics and matrix problems were anticipated but not yetclearlyunderstood.Nowtheserelationsarewellunderstood;theyhavebe- come precise statements rather than vague analogies. These relations are also considered in the lectures by E. Bresin and V. Kasakov which have appeared in the other volume of the School proceedings. From this point of view, four lectures by A. Okounkov take a particular position. They contain a sketch of the complete proof (obtained jointly with R.Pandharipande)oftheWitten–Kontsevichformularelatingthegenerating function of important combinatorial numbers of algebraic geometrical origin andtheτ-functionoftheKdVequationhierarchy.Intheselectures,specialat- tentionispaidtotheroleofthetheoryofsymmetricfunctionsandasymptotic representation theory, as well as to relations to random matrices. ThelecturesbyR.Speicher,M.Nazarov,andbyM.Boz˙ejkoandR.Szwarc aredevotedtomorespecialtopicswhich,however,fitinthesamecontext.At presenttherearehundredsofjournalpapersonallthesesubjects,howeverthe time for accomplished presentations is yet to come. The published lectures of theSchoolshouldstimulatethisprocess.Thereadershouldkeepinmindthat thereferencescitedinthelecturesarenotexhaustive.Ofcourse,therelations between asymptotic combinatorics and mathematical physics extend farther than the topics touched upon during the School. For example, closely related combinatorialproblemsplayakeyroleinconformalfieldtheorywhichisnow developingfast.Ihopethatthelecturespresentedinthisvolumewillbeuseful Preface vii forbeginnersaswellasforspecialistswhowanttofamiliarizethemselveswith this fascinating area of modern mathematics and mathematical physics and start working in it. Two prominent mathematicians, Anatoly Izergin (1948–1999) and Sergey Kerov(1946–2000),diedayearbeforetheconferencewhichhadbeenplanned with their active participation. Their contribution to areas of mathematical physicsandmathematicsrelatedtothetopicsoftheconferencewasenormous. One of the sessions of the conference was devoted to their memory. All the work on preparation of this manuscript was carried out by Yu. Yakubovich, to whom I am very grateful. I would like to express my gratitude to the following organizations which helped greatly in the organi- zation of the School: the European Mathematical Society, the NATO Science Committee, the Euler International Mathematical Institute, the St. Peters- burg Department of the Mathematical Institute of the Russian Academy of Sciences and the St. Petersburg Mathematical Society. Anatoly M. Vershik Contents Part I Random matrices, orthogonal polynomials and Riemann– Hilbert problem Asymptotic representation theory and Riemann–Hilbert problem A. Borodin...................................................... 3 Four Lectures on Random Matrix Theory P. Deift ........................................................ 21 Free Probability Theory and Random Matrices R. Speicher ..................................................... 53 Part II Algebraic geometry, symmetric functions and harmonic analysis A Noncommutative Version of Kerov’s Gaussian Limit for the Plancherel Measure of the Symmetric Group A. Hora ........................................................ 77 Random trees and moduli of curves A. Okounkov .................................................... 89 An introduction to harmonic analysis on the infinite symmetric group G. Olshanski ....................................................127 Two lectures on the asymptotic representation theory and statistics of Young diagrams A. Vershik ......................................................161 x Contents Part III Combinatorics and representation theory Characters of symmetric groups and free cumulants P. Biane........................................................185 Algebraic length and Poincar´e series on reflection groups with applications to representations theory M. Boz˙ejko and R. Szwarc ........................................201 Mixed hook-length formula for degenerate affine Hecke algebras M. Nazarov .....................................................223 Addendum Information about the school Information about the school ..................................239 Asymptotic representation theory and Riemann–Hilbert problem Alexei Borodin School of Mathematics Institute for Advanced Study Einstein Drive Princeton NJ 08540 U.S.A. [email protected] Summary. We show how the Riemann–Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptoticcombinatoricsandrepresentationtheory.TheWhittakerkernelandthe discrete Bessel kernel are computed as examples. Introduction A (discrete or continuous) random point process is called determinantal if its correlation functions have the form ρ (x ,...,x )=det[K(x ,x )]n , n 1 n i j i,j=1 where K(x,y) is a function in two variables called the correlation kernel. A major source of such point processes is Random Matrix Theory. All the “unitary” or “β = 2” ensembles of random matrices lead to determinantal point processes which describe the eigenvalues of these matrices. Determinantal point processes also arise naturally in problems of asymp- toticcombinatoricsandasymptoticrepresentationtheory,see[6]–[9],[5],[15], [21]. Usually, it is not very hard to see that the process that we are interested in is determinantal. A harder problem is to compute the correlation kernel of this process explicitly. The goal of this paper is to give an informal introduc- tion to a new method of obtaining explicit formulas for correlation kernels. It should be emphasized that in representation theoretic models which we consider the kernels cannot be expressed through orthogonal polynomials, as it often happens in random matrix models. That is why we had to invent something different. TheheartofthemethodistheRiemann–Hilbertproblem(RHP,forshort). Thisisaclassicalproblemwhichconsistsoffactorizingamatrix–valuedfunc- tion on a contour in the complex plane into a product of a function which A.Borodinetal.:LNM1817,A.M.Vershik(Ed.),pp.3–19,2003. (cid:1)c Springer-VerlagBerlinHeidelberg2003
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