Applications of Random Matrices in Physics

517 Pages·2006·3.28 MB·English

by  J. P. Keating (auth.), Édouard Brézin, Vladimir Kazakov, Didina Serban, Paul Wiegmann, Anton Zabrodin (eds.)

Applications of Random Matrices in Physics
NATO Science Series
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Series II:Mathematics,Physics and Chemistry – Vol.221
Applications of Random Matrices
in Physics
edited by
Édouard Bré zin
Laboratoire de Physique Théorique,
Ecole Normale Supérieure, Paris, France
Vladimir Kazakov
Laboratoire de Physique Théorique
,
de lEcole Normale Supérieure,
Université Paris-VI, Paris, France
Didina Serban
Service de Physique Théorique, CEA Saclay,
Gif-sur-Yvette Cedex, France
Paul Wiegmann
James Frank Institute,
University of Chicago, Chicago, IL, U.S.A.
and
Anton Zabrodin
Institute of Biochemical Physics, Moscow, Russia
and ITEP, Moscow, Russia
Proceedings of the NATO Advanced Study Institute on
Applications of RandomMatrices in Physics
Les Houches, France
6-25 June 2004
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-13 978-1-4020-4529-5 (HB)
ISBN-10 1-4020-4531-X (e-book)
ISBN-13 978-1-4020-4531-8 (e-book)
Published by Springer,
P.O.Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
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© 2006 Springer
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Contents
Preface ix
RandomMatricesandNumberTheory 1
J.P.Keating
1 Introduction 1
2 ζ(1 +it)andlogζ(1 +it) 9
2 2
3 Characteristicpolynomialsofrandomunitarymatrices 12
4 Othercompactgroups 17
5 FamiliesofL-functionsandsymmetry 19
6 Asymptoticexpansions 25
References 30
2DQuantumGravity,MatrixModelsandGraphCombinatorics 33
P.DiFrancesco
1 Introduction 33
2 Matrixmodelsfor2Dquantumgravity 35
3 Theone-matrixmodelI:largeN limitandtheenumerationofplanar
graphs 45
4 Thetreesbehindthegraphs 54
5 Theone-matrixmodelII:topologicalexpansionsandquantumgravity 58
6 Thecombinatoricsbeyondmatrixmodels:geodesicdistancein
planargraphs 69
7 Planargraphsasspatialbranchingprocesses 76
8 Conclusion 85
References 86
EigenvalueDynamics,FollytonsandLargeN LimitsofMatrices 89
JoakimArnlind,JensHoppe
References 93
RandomMatricesandSupersymmetryinDisorderedSystems 95
K.B.Efetov
1 Supersymmetrymethod 104
2 Wavefunctionsfluctuationsinafinitevolume.Multifractality 118
3 Recentandpossiblefuturedevelopments 126
4 Summary 134
Acknowledgements 134
References 134
vi APPLICATIONSOFRANDOMMATRICESINPHYSICS
HydrodynamicsofCorrelatedSystems 139
AlexanderG.Abanov
1 Introduction 139
2 Instantonorrarefluctuationmethod 142
3 Hydrodynamicapproach 143
4 Linearizedhydrodynamicsorbosonization 145
5 EFPthroughanasymptoticsofthesolution 147
6 Freefermions 148
7 Calogero-Sutherlandmodel 150
8 Freefermionsonthelattice 152
9 Conclusion 156
Acknowledgements 157
Appendix:Hydrodynamicapproachtonon-Galileaninvariantsystems 157
Appendix:ExactresultsforEFPinsomeintegrablemodels 158
References 160
QCD, ChiralRandomMatrixTheoryandIntegrability 163
J.J.M.Verbaarschot
1 Summary 163
2 Introduction 163
3 QCD 166
4 TheDiracspectruminQCD 174
5 LowenergylimitofQCD 176
6 ChiralRMTandtheQCDDiracspectrum 182
7 IntegrabilityandtheQCDpartitionfunction 188
8 QCDatfinitebaryondensity 200
9 FullQCDatnonzerochemicalpotential 211
10 Conclusions 212
Acknowledgements 213
References 214
EuclideanRandomMatrices:SolvedandOpenProblems 219
GiorgioParisi
1 Introduction 219
2 Basicdefinitions 222
3 Physicalmotivations 224
4 Fieldtheory 226
5 Thesimplestcase 230
6 Phonons 240
References 257
MatrixModelsandGrowthProcesses 261
A.Zabrodin
1 Introduction 261
2 Someensemblesofrandommatriceswithcomplexeigenvalues 264
Contents vii
3 ExactresultsatfiniteN 274
4 LargeN limit 282
5 Thematrixmodelasagrowthproblem 298
References 316
MatrixModelsandTopologicalStrings 319
~
MarcosMarino
1 Introduction 319
2 Matrixmodels 323
3 TypeBtopologicalstringsandmatrixmodels 345
4 TypeAtopologicalstrings,Chern-Simonstheoryandmatrixmodels 366
References 374
MatrixModelsofModuliSpace 379
SunilMukhi
1 Introduction 379
2 ModulispaceofRiemannsurfacesanditstopology 380
3 Quadraticdifferentialsandfatgraphs 383
4 ThePennermodel 388
5 Pennermodelandmatrixgammafunction 389
6 TheKontsevichModel 390
7 Applicationstostringtheory 394
8 Conclusions 398
References 400
MatrixModelsand2DStringTheory 403
EmilJ.Martinec
1 Introduction 403
2 Anoverviewofstringtheory 406
3 StringsinD-dimensionalspacetime 408
4 Discretizedsurfacesand2Dstringtheory 413
5 Anoverviewofobservables 421
6 Samplecalculation:thediskone-pointfunction 425
7 Worldsheetdescriptionofmatrixeigenvalues 434
8 Furtherresults 441
9 Openproblems 446
References 452
MatrixModelsasConformalFieldTheories 459
IvanK.Kostov
1 Introductionandhistoricalnotes 459
2 Hermitianmatrixintegral:saddlepointsandhyperellipticcurves 461
3 ThehermitianmatrixmodelasachiralCFT 470
4 Quasiclassicalexpansion:CFTonahyperellipticRiemannsurface 477
5 Generalizationtochainsofrandommatrices 483
References 486
viii APPLICATIONSOFRANDOMMATRICESINPHYSICS
Large N Asymptotics of Orthogonal Polynomials from Integrability to
AlgebraicGeometry 489
B.Eynard
1 Introduction 489
2 Definitions 489
3 Orthogonalpolynomials 490
4 Differentialequationsandintegrability 491
5 Riemann-Hilbertproblemsandisomonodromies 492
6 WKB–likeasymptoticsandspectralcurve 493
7 Orthogonalpolynomialsasmatrixintegrals 494
8 ComputationofderivativesofF(0) 495
9 Saddlepointmethod 496
10 Solutionofthesaddlepointequation 497
11 Asymptoticsoforthogonalpolynomials 507
12 Conclusion 511
References 511
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
MatrixTheoryhasgrownfarbeyondnuclear physics,andalsofarbeyondjust
levelstatistics. Itisconstantly developing intonewareasofphysicsandmath-
ematics, and now constitutes apart of the general culture and curriculum of a
theoretical physicist.
Mathematicalmethodsinspiredbyrandommatrixtheoryhavebecomepow-
erfulandsophisticated, andenjoyrapidlygrowinglistofapplications inseem-
inglydisconnected disciplines ofphysicsandmathematics.
A few recent, randomly ordered, examples of emergence of the Random
MatrixTheoryare:
- universalcorrelations inthemesoscopicsystems,
- disordered andquantum chaotic systems;
- asymptotic combinatorics;
- statistical mechanics onrandom planargraphs;
- problemsofnon-equilibriumdynamicsandhydrodynamics,growthmod-
els;
- dynamicalphasetransition inglasses;
- lowenergylimitsofQCD;
- advancesintwodimensionalquantumgravityandnon-criticalstringthe-
ory,areingreatpartduetoapplications oftheRandomMatrixTheory;
- superstring theoryandnon-abelian supersymmetric gaugetheories;
- zeros and value distributions of Riemann zeta-function, applications in
modularformsandellipticcurves;
- quantum andclassical integrable systemsandsolitontheory.
x APPLICATIONSOFRANDOMMATRICESINPHYSICS
InthesefieldstheRandomMatrixTheoryshedsanewlightonclassicalprob-
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. Oneimportantunifyingground,andalsoamathematicalbasisforthe
RandomMatrixTheory, istheconcept ofintegrability. Thisisdespite thefact
thatthetheorywasinvented todescribe randomness.
The main goal of the school was to accentuate fascinating links between
differentproblemsofphysicsandmathematics,wherethemethodsoftheRan-
domMatrixTheoryhavebeensuccessfully 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
theRandomMatrixUniverse.
Wearegratefully indebtedtoEldadBettelheimforhishelpinpreparingthe
volume.
EDITORS