Table Of ContentSPRINGER BRIEFS IN MATHEMATICAL PHYSICS 11
Makoto Katori
Bessel Processes,
Schramm–
Loewner
Evolution, and
the Dyson Model
123
SpringerBriefs in Mathematical Physics
Volume 11
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
More information about this series at http://www.springer.com/series/11953
Makoto Katori
Bessel Processes,
–
Schramm Loewner
Evolution, and the
Dyson Model
123
Makoto Katori
Department ofPhysics
ChuoUniversity
Tokyo
Japan
Additional material tothis bookcanbedownloaded from http://extras.springer.com.
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
ISBN978-981-10-0274-8 ISBN978-981-10-0275-5 (eBook)
DOI 10.1007/978-981-10-0275-5
LibraryofCongressControlNumber:2015959919
©TheAuthor(s)2015
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
foranyerrorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd.
To Hiroko, Machiko, and Rieko
Preface
Thisbookisbasedonmygraduate-courselecturesgivenattheGraduateSchoolof
Mathematics of the University of Tokyo in October 2008 (at the invitation of
T.FunakiandM.Jimbo),attheDepartmentofPhysicsoftheUniversityofTokyo
in November 2010 (at the invitation of S. Miyashita), at the Department of
MathematicsofTokyoInstituteofTechnologyinDecember2010(attheinvitation
ofK.Uchiyama),atÉcoledePhysiquedesHouches(LesHouchesPhysicsSchool)
in May 2011 (organized by C. Donati-Martin, S. Péché and G. Schehr), at the
Faculty of Mathematics of Kyushu University in June 2013 (at the invitation of
H. Osada and T. Shirai), and at the Graduate School of Arts and Sciences of the
University of Tokyo in July 2014 (at the invitation of A. Shimizu). First of all
I would like to thank those organizers for giving me such opportunities.
Thepurposeofmylecturesistointroducerecenttopicsinmathematicalphysics
and probability theory, especially the topics on the Schramm–Loewner evolution
(SLE) and interacting particle systems related to random matrix theory. A typical
exampleofthelattersystemsisDyson’sBrownianmotionmodel.Forthispurpose
IhaveconsideredonestorytotelltheSLEandtheDysonmodelas‘children’ofthe
Bessel processes. The Bessel processes make a one-parameter family of
one-dimensionaldiffusionprocesseswithparameterD,inwhichtheD-dimensional
Besselprocess,BES(D),isdefinedastheradialpartoftheD-dimensionalBrownian
motion, if D is a positive integer. This definition implies that Bessel processes are
‘children’ of the Brownian motion, and hence, the SLE and the Dyson model are
‘grandchildren’ of the Brownian motion.
The organization of this book is very simple. In Chap. 1 the parenthood of
Brownian motion in diffusion processes is clarified and we define BES(D) for any
D(cid:1)1. There, the importance of two aspects of BES(3) is explained. SLE is intro-
duced as a complexification of BES(D) in Chap. 2. We show that rich mathematics
andphysicsinvolvedinSLEareduetothenontrivialdependenceoftheBesselflow
on D. In Chap.3 Dyson’sBrownian motion model withparameter β is introduced
as a multivariate extension of BES(D) with the relation D¼βþ1. We will con-
centrate on the case where β¼2. In this case the Dyson model inherits the two
vii
viii Preface
aspects of BES(3) and has very strong solvability. That is, the process is proved to
bedeterminantalinthesensethatallspatio-temporalcorrelationfunctionsaregiven
by determinants, and all of them are controlled by a single function called the
correlation kernel.
Many parts of this book come from the joint work with Hideki Tanemura.
I thank him very much for the fruitful collaboration over 10 years. I would like to
thank Alexei Borodin, John Cardy, Patrik Ferrari, Peter John Forrester, Piotr
Graczyk, Kurt Johansson, Takashi Imamura, Christian Krattenthaler, Takashi
Kumagai, Neil O’Connell, Hirofumi Osada, Tomohiro Sasamoto, Grégory Schehr,
Tomoyuki Shirai, and Craig Tracy for giving me encouragement to prepare the
manuscript. I am grateful to Nizar Demni, Sergio Andraus, Syota Esaki, Ryoki
Fukushima, andShuta Nakajima for careful readingofthedraftandalot ofuseful
comments. All suggestions given by two anonymous reviewers of this book are
very important and useful for improving the text and I acknowledge their efforts
very much. Thanks are due to Naoki Kobayashi and Kan Takahashi for preparing
several figures in the book.
I thank Masayuki Nakamura at the Editorial Department of Springer Japan for
his truly kind assistance during the preparation of this manuscript.
The research of the author was supported in part by the Grant-in-Aid for
Scientific Research (C) (No.21540397 and No.26400405) of the Japan Society for
the Promotion of Science.
Tokyo Makoto Katori
December 2015
Contents
1 Bessel Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 One-Dimensional Brownian Motion (BM). . . . . . . . . . . . . . . . 1
1.2 Martingale Polynomials of BM . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Drift Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Stochastic Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Itô’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Complex Brownian Motion and Conformal Invariance . . . . . . . 16
1.8 Stochastic Differential Equations for Bessel Processes. . . . . . . . 17
1.9 Kolmogorov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.10 BESð3Þ and Absorbing BM . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.11 BESð1Þ and Reflecting BM . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.12 Critical Dimension Dc ¼2 . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.13 Bessel Flow and Another Critical Dimension Dc ¼3=2 . . . . . . 28
1.14 Hypergeometric Functions Representing Bessel Flow. . . . . . . . 32
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Schramm–Loewner Evolution (SLE). . . . . . . . . . . . . . . . . . . . . . . 41
2.1 Complexification of Bessel Flow . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Schwarz–Christoffel Formula and Loewner Chain . . . . . . . . . . 45
2.3 Three Phases of SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4 Cardy’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 SLE and Statistical Mechanics Models . . . . . . . . . . . . . . . . . . 54
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Dyson Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Multivariate Extension of Bessel Process . . . . . . . . . . . . . . . . 57
3.2 Dyson Model as Eigenvalue Process . . . . . . . . . . . . . . . . . . . 59
ix
x Contents
3.3 Dyson Model as Noncolliding Brownian Motion . . . . . . . . . . . 65
3.4 Determinantal Martingale Representation (DMR). . . . . . . . . . . 72
3.5 Reducibility of DMR and Correlation Functions . . . . . . . . . . . 77
3.5.1 Density Function ρξðt;xÞ. . . . . . . . . . . . . . . . . . . . . . 79
3.5.2 Two-Time Correlation Function ρξðs;x;t;yÞ . . . . . . . . 80
3.6 Determinantal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Constant-Drift Transformation of Dyson Model. . . . . . . . . . . . 86
3.8 Generalization for Initial Configuration with Multiple Points. . . 89
3.9 Wigner’s Semicircle Law and Scaling Limits . . . . . . . . . . . . . 93
3.9.1 Wigner’s Semicircle Law . . . . . . . . . . . . . . . . . . . . . 93
3.9.2 Bulk Scaling Limit and Homogeneous
Infinite System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.9.3 Soft-Edge Scaling Limit and Spatially Inhomogeneous
Infinite System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.10 Entire Functions and Infinite Particle Systems . . . . . . . . . . . . . 99
3.10.1 Nonequilibrium Sine Process. . . . . . . . . . . . . . . . . . . 101
3.10.2 Nonequilibrium Airy Process. . . . . . . . . . . . . . . . . . . 105
3.11 Tracy–Widom Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.11.1 Distribution Function of the Maximum Position of
Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.11.2 Integrals Involving Resolvent of Correlation Kernel . . . 114
3.11.3 Nonlinear Third-Order Differential Equation . . . . . . . . 115
3.11.4 Soft-Edge Scaling Limit . . . . . . . . . . . . . . . . . . . . . . 117
3.11.5 PainlevéII and Limit Theorem of Tracy and Widom. . .. 118
3.12 Beyond Determinantal Processes . . . . . . . . . . . . . . . . . . . . . . 122
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
