Applications of Random Matrices in Physics NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Springer Sub-Series I. Life and Behavioural Sciences IOS Press II. Mathematics, Physics and Chemistry III. Computer and Systems Science IOS Press IV. Earth and Environmental Sciences The NATO Science Series continues the series of books published formerly as the NATO ASI Series. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series was re-organized to the four sub-series noted above. Please consult the following web sites for information on previous volumes published in the Series. http://www .nato .int/science http://www .iospress .nl – The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, and the NATO Science Series collects together the results of these meetings. The meetings are co-organized by scientists from NATO countries and scientists from NAT O s Partner countries countries of the CIS and Central and Eastern Europe. , in conjunction with the NATO Public Diplomacy Division. Springer Springer http://www.springer.com Series II: Mathematics, Physics and Chemistry – Vol. 221 Applications of Random Matrices in Physics edited by zin Vladimir Kazakov Universit Paris-VI, Paris, France Didina Serban Service de Physique Th orique, CEA Saclay, Gif-sur-Yvette Cedex, France Paul Wiegmann and Anton Zabrodin Institute of Biochemical Physics, Moscow, Russia É é é and ITEP, Moscow, Russia Ecole Normale Sup ri ure, Paris, France é ThLaboratoire de Physique orique, é ThLaboratoire de Physique orique , douard Br de l Ecole Normale Sup rieure, é University of Chicago, Chicago, IL, U.S.A. James Frank Institute, é é e A C.I.P.Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4530-1 (PB) ISBN-13 978-1-4020-4530-1 (PB) ISBN-10 1-4020-4529-8 (HB) ISBN-10 1-4020-4531-X (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands. Proceedings of the NATO Advanced Study Institute on Applications of Random ISBN-13 978-1-4020-4531-8 (e-book) ISBN-13 978-1-4020-4529-5 (HB) 6 - 25 June 2004 Les Houches, France Matrices in Physics Contents Preface ix 1 J. P. Keating 1 Introduction 1 2 ζ( 1 2 + it) and log ζ( 1 2 + it) 9 3 Characteristic polynomials of random unitary matrices 12 417 5 19 6 Asymptotic expansions 25 References 30 2D Quantum Gravity, Matrix Models and Graph Combinatorics 33 P. Di Francesco 1 Introduction 33 2 Matrix models for 2D quantum gravity 35 3 The one-matrix model I: large N limit and the enumeration of planar graphs 45 4 The trees behind the graphs 54 5 The one-matrix model II: topological expansions and quantum gravity 6 The combinatorics beyond matrix models: geodesic distance in pla nar graphs 7 Planar graphs as spatial branching processes 8 Conclusion References Joakim Arnlind, Jens Hoppe References K.B. Efetov 1 Supersymmetry method 2 Wave functions fluctuations in a finite volume. Multifractality 3 Recent and possible future developments 4 Summary Random Matrices and Number Theory Other compact groups Eigenvalue Dynamics, Follytons and Large N Limits of Matrices References 58 69 76 85 86 89 93 95 104 118 126 134 134 134 Families of L-functions and symmetry Acknowledgements Random Matrices and Supersymmetry in Disordered Systems vi APPLICATIONS OF RANDOM MATRICES IN PHYSICS Alexander G. Abanov 1 Introduction 2 Instanton or rare fluctuation method 3 Hydrodynamic approach 4 Linearized hydrodynamics or bosonization 5 EFP through an asymptotics of the solution 9 Conclusion Appendix: Hydrodynamic approach to non-Galilean invariant systems Appendix: Exact results for EFP in some integrable models References J.J.M. Verbaarschot 1 Summary 2 Introduction 3QCD 4 5 6 7 8 9 10 Conclusions References Giorgio Parisi 1 Introduction 2 Basic definitions 3 Physical motivations 4 Field theory 5 The simplest case 6 Phonons Hydrodynamics of Correlated Systems QCD, Chiral Random Matrix Theory and Integrability Euclidean Random Matrices: Solved and Open Problems The Dirac spectrum in QCD Low energy limit of QCD Integrability and the QCD partition function Chiral RMT and the QCD Dirac spectrum QCD at finite baryon density Full QCD at nonzero chemical potential References 139 139 142 143 147 145 6 Free fermions 148 7 Calogero-Sutherland model 150 8 Free fermions on the lattice 152 157 156 157 158 160 163 163 163 166 174 176 182 188 200 211 212 213 219 219 222 224 226 230 240 257 A. Zabrodin 1 Introduction 2 Some ensembles of random matrices with complex eigenvalues Matrix Models and Growth Processes 261 261 264 214 Acknowledgements Acknowledgements Contents vii 3 Exact results at finite N 4LargeN limit 5 The matrix model as a growth problem References Marcos Mari ~ no 1 Introduction 2 Matrix models 3 Type B topological strings and matrix models 4 Type A topological strings, Chern-Simons theory and matrix models References Matrix Models of Moduli Space Sunil Mukhi 1 Introduction 2 3 4 The Penner model 5 6 The Kontsevich Model 7 8 Conclusions References Matrix Models and 2D String Theory Emil J. Martinec 1 Introduction 2 An overview of string theory 3 Strings in D-dimensional spacetime 4 Discretized surfaces and 2D string theory 5 An overview of observables 6 Sample calculation: the disk one-point function 7 Worldsheet description of matrix eigenvalues 8 Further results 9 Open problems Matrix Models and Topological Strings Quadratic differentials and fatgraphs Penner model and matrix gamma function Applications to string theory References 274 282 298 316 319 319 323 345 366 374 379 379 380 383 388 389 390 394 398 400 403 403 408 413 421 425 434 441 406 446 Matrix Models as Conformal Field Theories Ivan K. Kostov 1 Introduction and historical notes 2 Hermitian matrix integral: saddle points and hyperelliptic curves 3 The hermitian matrix model as a chiral CFT 4 Quasiclassical expansion: CFT on a hyperelliptic Riemann surface 5 Generalization to chains of random matrices References 459 459 461 470 477 483 486 Moduli space of Riemann surfaces and its topology 452 B. Eynard 1 Introduction 2 Definitions 3 Orthogonal polynomials 4 5 Riemann-Hilbert problems and isomonodromies 6 WKB–like asymptotics and spectral curve 7 Orthogonal polynomials as matrix integrals 8 (0) 9 10 Solution of the saddlepoint equation 11 Asymptotics of orthogonal polynomials 12 Conclusion References 489 489 489 490 Differential equations and integrability 491 492 493 494 495 496 497 507 511 viii APPLICATIONS OF RANDOM MATRICES IN PHYSICS Saddle point method 511 Large N Asymptotics of Orthogonal Polynomials from Integrability to Algebraic Computation of derivatives of F Geometry Preface Random matrices are widely and successfully used in physics for almost 60-70 years, beginning with the works of Wigner and Dyson. Initially pro- posed to describe statistics of excited levels in complex nuclei, the Random Matrix Theory has grown far beyond nuclear physics, and also far beyond just level statistics. It is constantly developing into new areas of physics and math- ematics, and now constitutes a part of the general culture and curriculum of a theoretical physicist. Mathematical methods inspired by random matrix theory have become pow- erful and sophisticated, and enjoy rapidly growing list of applications in seem- ingly disconnected disciplines of physics and mathematics. A few recent, randomly ordered, examples of emergence of the Random Matrix Theory are: - universal correlations in the mesoscopic systems, - disordered and quantum chaotic systems; - asymptotic combinatorics; - statistical mechanics on random planar graphs; - problems of non-equilibrium dynamics and hydrodynamics, growth mod- els; - dynamical phase transition in glasses; - low energy limits of QCD; - advances in two dimensional quantum gravity and non-critical string the- ory, are in great part due to applications of the Random Matrix Theory; - superstring theory and non-abelian supersymmetric gauge theories; - zeros and value distributions of Riemann zeta-function, applications in modular forms and elliptic curves; - quantum and classical integrable systems and soliton theory. x APPLICATIONS OF RANDOM MATRICES IN PHYSICS In these fields the Random Matrix Theory sheds a new light on classical prob- lems. On the surface, these subjects seem to have little in common. In depth the subjects are related by an intrinsic logic and unifying methods of theoretical physics. One important unifying ground, and also a mathematical basis for the Random Matrix Theory, is the concept of integrability. This is despite the fact that the theory was invented to describe randomness. The main goal of the school was to accentuate fascinating links between different problems of physics and mathematics, where the methods of the Ran- dom Matrix Theory have been successfully used. We hope that the current volume serves this goal. Comprehensive lectures and lecture notes of seminars presented by the leading researchers bring a reader to frontiers of a broad range of subjects, applications, and methods of the Random Matrix Universe. We are gratefully indebted to Eldad Bettelheim for his help in preparing the volume. EDITORS [...]... distribution of c|d| should be related to that described in (70) 5.4 Frequency of vanishing of L-functions I now turn to the question of the frequency of vanishing of L-functions at the central point In the light of the Birch & Swinnerton-Dyer conjecture, which relates the order of vanishing at this point to the number of rational points on the corresponding elliptic curve, this is an issue of considerable... term coming from multiplying the bottom-left entry by the top-right entry and all of the diagonal entries on the other rows Thus the combined diagonal and off-diagonal terms add up to give the expression in (16), bearing in mind that when k = 0 the total is just N 2 , the number of terms in the sum over j and l 6 APPLICATIONS OF RANDOM MATRICES IN PHYSICS Heine’s identity itself may be proved using the... −6, etc., and in nitely many zeros, called the non-trivial zeros, 1 E Brezin et al (eds.), Applications of Random Matrices in Physics, 1–32 © 2006 Springer Printed in the Netherlands 2 APPLICATIONS OF RANDOM MATRICES IN PHYSICS in the critical strip 0 < Res < 1 It satisfies the functional equation π −s/2 Γ s ζ(s) = π −(1−s)/2 Γ 2 1−s 2 ζ(1 − s) (2) The Riemann Hypothesis states that all of the non-trivial... distributions of the zeros of the Riemann zeta function and other L-functions and those of the eigenvalues of random matrices associated with the classical compact groups, on the scale of the mean zero/eigenvalue spac- 10 APPLICATIONS OF RANDOM MATRICES IN PHYSICS ing My goal in the remainder of these notes is to focus on more recent developments that concern the value distribution of the functions ζ( 1... distribution 2 of L-functions within families, and applications of these results to some other important questions in number theory The basic ideas I shall be reviewing were introduced in [24], [25], and [7] The theory was substantially developed in [8, 9] The applications I shall describe later were initiated in [12] and [13] Details of all of the calculations I shall outline can be found in these references... functions form a family of L-functions parameterized by the integer index d (This is the family mentioned in the Introduction.) 20 APPLICATIONS OF RANDOM MATRICES IN PHYSICS 5.2 Example 2: L-functions associated with elliptic curves Consider the function ∞ f (z) = e2πiz (1 − e2πinz )2 (1 − e22πinz )2 n=1 ∞ an e2πinz , = (73) n=1 where the integers an are the Fourier coefficients of f This function may... as the matrix size tends to in nity It is important to note that the proof of Montgomery’s theorem does not involve any of the steps in the derivation of the CUE pair correlation function It is instead based entirely on the connection between the Riemann zeros and the primes In outline, the proof involves computing the pair correlation function of the derivative of N (T ) Using the explicit formula (5),... most of the interval in question The flatness observed therefore reflects the main dependence on D3/4 (From [12].) Data in support of the second conjecture are listed in Table 2 and are plotted in Figure 6 In this case the agreement with the conjecture is striking 6 Asymptotic expansions The limit (30) may be thought of as representing the leading-order asymptotics of the moments of the zeta function, in. .. deviations from this simple-minded ansatz Assuming that the mo2 T ments do indeed grow like (log 2π )λ , the problem is then to determine fζ (λ) 12 APPLICATIONS OF RANDOM MATRICES IN PHYSICS CUE 15 Zeta Gaussian 10 5 -6 -4 -2 2 4 Figure 2 The logarithm of the inverse of the value distribution plotted in Figure 1 (Taken from [24].) The conjecture is known to be correct in only two non-trivial cases,... not time-reversal invariant is consistent with the conjecture that the energy level statistics of such systems should, generically, in the semiclassical limit, coincide with those of the eigenvalues of random matrices from one of the ensembles that are invariant under unitary transformations, such as the CUE or the GUE, in the limit of large matrix size [4] The appearance of random matrices associated . Brezin et al. (eds.), Applications of Random Matrices in Physics, 1–32. © 2006 Springer. Printed in the Netherlands. 1 APPLICATIONS OF RANDOM MATRICES IN. growing list of applications in seem- ingly disconnected disciplines of physics and mathematics. A few recent, randomly ordered, examples of emergence of