本书为“国外物理名著系列”丛书之一，系统阐述了随机矩阵的理论知识。适合对随机矩阵处理物理问题感兴趣的研究生和科研人员参考。

随机矩阵理论相关的数学方法能够解决更多的问题，而且方式更加灵活，在物理学中的应用也更加深入，可以用来计算介观系统的通用关系。它在无序系统和量子混沌领域也有一些新的应用，并且通过建立新的矩阵模型，在二维引力和弦以及非阿贝尔规范理论方面取得了重要进展。

Dyson和Wigner最先成功地将随机矩阵应用到物理学中，经过六七十年的发展，现在它在物理学中的应用越来越广泛，并且已经渗透到了现代数学、物理学的很多新兴领域，是理论物理学家的重要数学工具。随机矩阵理论相关的数学方法能够解决更多的问题，而且方式更加灵活，在物理学中的应用也更加深入，可以用来计算介观系统的通用关系。它在无序系统和量子混沌领域也有一些新的应用，并且通过建立新的矩阵模型，在二维引力和弦以及非阿贝尔规范理论方面取得了重要进展。

本书由本领域的杰出学者撰写，系统阐述了相关的理论知识。适合对随机矩阵处理物理问题感兴趣的研究生和科研人员参考。

Preface

Random Matrices and Number Theory

J.P. Keating

1 Introduction

2 ζ(1/2+it)and logζ(1/2+it)

3 Characteristic polynomials of random unitary matrices

4 Other compact groups

5 Families of L-functions and symmetry

6 Asymptotic expansions

References

2D Quantum Gravity, Matrix Models and Graph Combinatorics

P. Di Francesco

1 Introduction

2 Matrix models for 2D quantum gravity

3 The one-matrix model I: large N limit and the enumeration of planar graphs

4 The trees behind the graphs

5 The one-matrix model II：topological expansions and quantum gravity 58

6 The combinatorics beyond matrix models: geodesic distance in planar graphs

7 Planar graphs as spatial branching processes

8 Conclusion

References

Eigenvalue Dynamics, Follytons and Large N Limits of Matrices

Joakim Arnlind, Jens Hoppe

References

Random Matrices and Supersymmetry in Disordered Systems

K.B. Efetov

1 Supersymmetry method

2 Wave functions fluctuations in a finite volume. Multifractality

3 Recent and possible future developments

4 Summary

Acknowledgements

References

Hydrodynamics of Correlated Systems

Alexander G．Abanoy

1 Introduction

2 Instanton or rare fluctuation method

3 Hydrodynam ic approach

4 Linearized hydrodynamics or bosoflization

5 EFP through an asymptotics of the solution

6 Free fermions

7 Calogero-Sutherland model

8 Free fermions on the lattice

9 Conclusion

Acknowledgements

Appendix：Hydrodynamic approach to non-Galilean invariant systems

Appendix：Exact results for EFP in some integrable models

References

QCD，Chiral Random Matrix Theory and Integrability

J．JM．Verbaarschot

1 Summarv

2 IntrodUCtion

3 OCD

4 The Dirac spectrum in QCD

5 Low eflergy limit of QCD

6 Chiral RMT and the QCD Dirac spectrum

7 Integrability and the QCD partition function

8 QCD at fin ite baryon density

9 Full QCD at nonzero chemical potential

10 Conclusions

Acknowledgements

References

EUClidean Random Matrices：SOlved and Open Problems

Giorgio Parisi

1 Introduction

2 Basic definitions

3 Physical motivations

4 Field theory

5 The simplest case

6 Phonons

References

Matrix Models and Growth Processes3

A．Zabrodin

1 Introduction

2 Some ensembles of random matrices with cornplex eigenvalues

3 Exact results at finite N

4 Large N limit

5 The matrix model as a growth problem

References

Matrix Models and Topological Strings

Marcos Marino

1 Introduction

2 Matrix models

3 Type B topological strings and matrix models

4 Type A topological strings, Chern-Simons theory and matrix models 366

References

Matrix Models of Moduli Space

Sunil Mukhi

1 Introduction

2 Moduli space of Riemann surfaces and its topology

3 Quadratic differentials and fatgraphs

4 The Penner model

5 Penner model and matrix gamma function

6 The Kontsevich Model

7 Applications to string theory

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

References

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

Large N Asymptotics of Orthogonal Polynomials from Integrability to Algebraic Geometry

B. Eynard

1 Introduction

2 Definitions

3 Orthogonal polynomials

4 Differential equations and integrability

5 Riemann-Hilbert problems and isomonodromies

6 WKB-like asymptotics and spectral curve

7 Orthogonal polynomials as matrix integrals

8 Computation of derivatives of F(0)

9 Saddle point method

10 Solution of the saddlepoint equation

11 Asymptotics of orthogonal polynomials

12 Conclusion

References