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Applications of Random Matrices in Physics
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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
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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
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