随机矩阵在物理学中的应用

Preface

RandomMatricesandNumberTheory

J.P.Keating

1Introduction

2ζ(1/2+it)andlogζ(1/2+it)

3Characteristicpolynomialsofrandomunitarymatrices

4Othercompactgroups

5FamiliesofL-functionsandsymmetry

6Asymptoticexpansions

References

2DQuantumGravity,MatrixModelsandGraphCombinatorics

P.DiFrancesco

1Introduction

2Matrixmodelsfor2Dquantumgravity

3Theone-matrixmodelI:largeNlimitandtheenumerationofplanargraphs

4Thetreesbehindthegraphs

5Theone-matrixmodelII：topologicalexpansionsandquantumgravity58

6Thecombinatoricsbeyondmatrixmodels:geodesicdistanceinplanargraphs

7Planargraphsasspatialbranchingprocesses

8Conclusion

References

EigenvalueDynamics,FollytonsandLargeNLimitsofMatrices

JoakimArnlind,JensHoppe

References

RandomMatricesandSupersymmetryinDisorderedSystems

K.B.Efetov

1Supersymmetrymethod

2Wavefunctionsfluctuationsinafinitevolume.Multifractality

3Recentandpossiblefuturedevelopments

4Summary

Acknowledgements

References

HydrodynamicsofCorrelatedSystems

AlexanderG．Abanoy

1Introduction

2Instantonorrarefluctuationmethod

3Hydrodynamicapproach

4Linearizedhydrodynamicsorbosoflization

5EFPthroughanasymptoticsofthesolution

6Freefermions

7Calogero-Sutherlandmodel

8Freefermionsonthelattice

9Conclusion

Acknowledgements

Appendix：Hydrodynamicapproachtonon-Galileaninvariantsystems

Appendix：ExactresultsforEFPinsomeintegrablemodels

References

QCD，ChiralRandomMatrixTheoryandIntegrability

J．JM．Verbaarschot

1Summarv

2IntrodUCtion

3OCD

4TheDiracspectruminQCD

5LoweflergylimitofQCD

6ChiralRMTandtheQCDDiracspectrum

7IntegrabilityandtheQCDpartitionfunction

8QCDatfinitebaryondensity

9FullQCDatnonzerochemicalpotential

10Conclusions

Acknowledgements

References

EUClideanRandomMatrices：SOlvedandOpenProblems

GiorgioParisi

1Introduction

2Basicdefinitions

3Physicalmotivations

4Fieldtheory

5Thesimplestcase

6Phonons

References

MatrixModelsandGrowthProcesses3

A．Zabrodin

1Introduction

2Someensemblesofrandommatriceswithcornplexeigenvalues

3ExactresultsatfiniteN

4LargeNlimit

5Thematrixmodelasagrowthproblem

References

MatrixModelsandTopologicalStrings

MarcosMarino

1Introduction

2Matrixmodels

3TypeBtopologicalstringsandmatrixmodels

4TypeAtopologicalstrings,Chern-Simonstheoryandmatrixmodels366

References

MatrixModelsofModuliSpace

SunilMukhi

1Introduction

2ModulispaceofRiemannsurfacesanditstopology

3Quadraticdifferentialsandfatgraphs

4ThePennermodel

5Pennermodelandmatrixgammafunction

6TheKontsevichModel

7Applicationstostringtheory

8Conclusions

References

MatrixModelsand2DStringTheory

EmilJ.Martinec

1Introduction

2Anoverviewofstringtheory

3StringsinD-dimensionalspacetime

4Discretizedsurfacesand2Dstringtheory

5Anoverviewofobservables

6Samplecalculation:thediskone-pointfunction

7Worldsheetdescriptionofmatrixeigenvalues

8Furtherresults

9Openproblems

References

MatrixModelsasConformalFieldTheories

IvanK.Kostov

1Introductionandhistoricalnotes

2Hermitianmatrixintegral:saddlepointsandhyperellipticcurves

3ThehermitianmatrixmodelasachiralCFT

4Quasiclassicalexpansion:CFTonahyperellipticRiemannsurface

5Generalizationtochainsofrandommatrices

References

LargeNAsymptoticsofOrthogonalPolynomialsfromIntegrabilitytoAlgebraicGeometry

B.Eynard

1Introduction

2Definitions

3Orthogonalpolynomials

4Differentialequationsandintegrability

5Riemann-Hilbertproblemsandisomonodromies

6WKB-likeasymptoticsandspectralcurve

7Orthogonalpolynomialsasmatrixintegrals

8ComputationofderivativesofF(0)

9Saddlepointmethod

10Solutionofthesaddlepointequation

11Asymptoticsoforthogonalpolynomials

12Conclusion

References