| 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。 |
| 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 |
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