Random matrices.

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.

[1]  J. Wishart THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[2]  F. Dyson The Dynamics of a Disordered Linear Chain , 1953 .

[3]  Harish-Chandra Differential Operators on a Semisimple Lie Algebra , 1957 .

[4]  E. Wigner On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[5]  H. Furstenberg,et al.  Products of Random Matrices , 1960 .

[6]  F. Dyson Statistical Theory of the Energy Levels of Complex Systems. I , 1962 .

[7]  Two Folk Lemmas on the Expansion of the S Matrix or the Out Field in Normal Ordered In Fields , 1962 .

[8]  F. Dyson A Brownian‐Motion Model for the Eigenvalues of a Random Matrix , 1962 .

[9]  N. R. Goodman The Distribution of the Determinant of a Complex Wishart Distributed Matrix , 1963 .

[10]  H. Furstenberg Noncommuting random products , 1963 .

[11]  M Rosenblatt,et al.  Products of independent identically distributed stochastic matrices , 1965 .

[12]  I. Lifshitz,et al.  Reviews of Topical Problems: Energy Spectrum Structure and Quantum States of Disordered Condensed Systems , 1965 .

[13]  E. Wigner Random Matrices in Physics , 1967 .

[14]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[15]  On the Fourier series of a stationary process. II , 1969 .

[16]  John D. Fay Theta Functions on Riemann Surfaces , 1973 .

[17]  L. Pastur Spectra of Random Self Adjoint Operators , 1973 .

[18]  F. Berezin Some remarks on the wigner distribution , 1973 .

[19]  J. Kingman Subadditive Ergodic Theory , 1973 .

[20]  G. Hooft A Planar Diagram Theory for Strong Interactions , 1974 .

[21]  A. K. Grincevicjus On the Continuity of the Distribution of a Sum of Dependent Variables Connected with Independent Walks on Lines , 1974 .

[22]  David Handelman,et al.  Rank functions and K0 of regular rings , 1976 .

[23]  M. Berry,et al.  Level clustering in the regular spectrum , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[24]  On the limit of the convolution iterates of a probability measure on n × n stochastic matrices , 1977 .

[25]  G. Parisi,et al.  Planar diagrams , 1978 .

[26]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[27]  Arunava Mukherjea,et al.  Limit Theorems: Stochastic Matrices, Ergodic Markov Chains, and Measures on Semigroups* , 1979 .

[28]  Limit behavior of the convolution iterates of a probability measure on a semigroup of matrices , 1979 .

[29]  Shripad Tuljapurkar,et al.  Population dynamics in variable environments I. Long-run growth rates and extinction , 1980 .

[30]  C. Itzykson The planar approximation , 1980 .

[31]  C. Itzykson,et al.  The planar approximation. II , 1980 .

[32]  A. Mukherjea,et al.  Recurrent random walks and invariant measures on semigroups ofn×n matrices , 1980 .

[33]  S. Geman A Limit Theorem for the Norm of Random Matrices , 1980 .

[34]  P. A. Mello,et al.  Random matrix physics: Spectrum and strength fluctuations , 1981 .

[35]  P. Krishnaiah,et al.  Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations , 1982 .

[36]  J. Duistermaat,et al.  On the variation in the cohomology of the symplectic form of the reduced phase space , 1982 .

[37]  A. Migdal Loop equations and 1N expansion , 1983 .

[38]  M. Jimbo,et al.  Solitons and Infinite Dimensional Lie Algebras , 1983 .

[39]  H. Helson Harmonic Analysis , 1983 .

[40]  Limit Theorems for Products of Independent Random Matrices with Positive Elements , 1983 .

[41]  O. Bohigas,et al.  Characterization of chaotic quantum spectra and universality of level fluctuation laws , 1984 .

[42]  Charles M. Newman,et al.  The Stability of Large Random Matrices and Their Products , 1984 .

[43]  K. Takasaki,et al.  Toda lattice hierarchy , 1984 .

[44]  F. Spitzer,et al.  Convergence in distribution of products of random matrices , 1984 .

[45]  V. Dotsenko,et al.  Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge C≤1 , 1984 .

[46]  S. Pincus Strong laws of large numbers for products of random matrices , 1985 .

[47]  P. Gallagher Pair correlation of zeros of the zeta function. , 1985 .

[48]  A. Raugi,et al.  Frontière de furstenberg, propriétés de contraction et théorèmes de convergence , 1985 .

[49]  C. Heyde,et al.  Confidence intervals for demographic projections based on products of random matrices. , 1985, Theoretical population biology.

[50]  Alexander A. Migdal,et al.  Ising model of a randomly triangulated random surface as a definition of fermionic string theory , 1986 .

[51]  Zhang,et al.  Dynamic scaling of growing interfaces. , 1986, Physical review letters.

[52]  Vladimir Kazakov,et al.  Ising model on a dynamical planar random lattice: Exact solution , 1986 .

[53]  A. Odlyzko On the distribution of spacings between zeros of the zeta function , 1987 .

[54]  P. Bougerol Tightness of Products of Random Matrices and Stability of Linear Stochastic Systems , 1987 .

[55]  G. Högnäs A note on products of random matrices , 1987 .

[56]  Eric S. Key,et al.  Computable examples of the maximal Lyapunov exponent , 1987 .

[57]  A. Mukherjea Convergence in distribution of products of random matrices: a semigroup approach , 1987 .

[58]  J. Elton An ergodic theorem for iterated maps , 1987, Ergodic Theory and Dynamical Systems.

[59]  A. Mukherjea,et al.  Stochastic flows on a countable set: Convergence in distribution , 1988 .

[60]  Robert C. Penner,et al.  Perturbative series and the moduli space of Riemann surfaces , 1988 .

[61]  I Ya Gol'dsheid,et al.  Lyapunov indices of a product of random matrices , 1989 .

[62]  M. Gaudin,et al.  O(n) model on a fluctuating planar lattice. Some exact results , 1989 .

[63]  I. Kostov O($n$) Vector Model on a Planar Random Lattice: Spectrum of Anomalous Dimensions , 1989 .

[64]  The Popularization of Mathematics: Mathematics and the Media , 1990 .

[65]  Vi︠a︡cheslav Leonidovich Girko,et al.  Theory of random determinants , 1990 .

[66]  E. S. Key Lower bounds for the maximal Lyapunov exponent , 1990 .

[67]  Matrix models of 2d gravity , 1991, hep-th/9112013.

[68]  A. Mukherjea,et al.  Convergence in distribution of products of d × d random matrices , 1991 .

[69]  Semigroups, Attractors, and Products of Random Matrices , 1991 .

[70]  R. W. R. Darling,et al.  Discrete Time Voter Models: A Class of Stochastic Automata , 1991 .

[71]  A. Mukherjea Tightness of products of i.i.d. random matrices , 1991 .

[72]  Maxim Kontsevich,et al.  Intersection theory on the moduli space of curves and the matrix airy function , 1992 .

[73]  A. Morozov PAIR CORRELATOR IN THE ITZYKSON-ZUBER INTEGRAL , 1992 .

[74]  A. Mukherjea Recurrent random walks in nonnegative matrices: attractors of certain iterated function systems , 1992 .

[75]  Eric Kostlan,et al.  On the spectra of Gaussian matrices , 1992 .

[76]  Convergence in Distribution of a Markov Process Generated by I.I.D. Random Matrices , 1992 .

[77]  Recurrent random walks in nonnegative matrices II , 1993 .

[78]  J. Karwowski Statistical theory of spectra , 1994 .

[79]  A. Edelman,et al.  How many eigenvalues of a random matrix are real , 1994 .

[80]  A. Barabasi,et al.  Fractal concepts in surface growth , 1995 .

[81]  A mixed random walk on nonnegative matrices: A law of large numbers , 1995 .

[82]  Exact solution of the O(n) model on a random lattice , 1995, hep-th/9506193.

[83]  E. Kirchberg On Subalgebras of the Car-Algebra , 1995 .

[84]  C. Tracy,et al.  Mathematical Physics © Springer-Verlag 1996 On Orthogonal and Symplectic Matrix Ensembles , 1995 .

[85]  L. Pastur Spectral and probabilistic aspects of matrix models , 1996 .

[86]  V. Dotsenko,et al.  Conformal Algebra and Multipoint Correlation Functions in 2d Statistical Models , 1996 .

[87]  Matthias Staudacher,et al.  Character expansion methods for matrix models of dually weighted graphs , 1995, hep-th/9502132.

[88]  C. Beenakker Book reviewSupersymmetry in disorder and chaos: by K. Efetov Cambridge University Press, 1997. £65.00 hbk (xiii + 441 pages) ISBN 0 521 47097 8 , 1997 .

[89]  H. C. Wu,et al.  Nuclear Data for Science and Technology , 1998 .

[90]  Nicholas M. Katz,et al.  Random matrices, Frobenius eigenvalues, and monodromy , 1998 .

[91]  C. Kristjansen,et al.  An iterative solution of the three-colour problem on a random lattice , 1998 .

[92]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[93]  K. Johansson On fluctuations of eigenvalues of random Hermitian matrices , 1998 .

[94]  M. L. Mehta,et al.  Matrices coupled in a chain: I. Eigenvalue correlations , 1998 .

[95]  Smoothed Correlators For Symmetric Double-Well Matrix Models: Some Puzzles and Resolutions , 1998, cond-mat/9805096.

[96]  Kelly Morris It's only a game , 1999, The Lancet.

[97]  P. Diaconis,et al.  Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem , 1999 .

[98]  É. Brézin,et al.  Correlations and symmetry breaking in gapped matrix models , 1999 .

[99]  L. Pastur RANDOM MATRICES AS PARADIGM , 2000 .

[100]  Ivan Kostov Exact solution of the six-vertex model on a random lattice , 2000 .

[101]  I. Krichever,et al.  Tau-function for Analytic Curves , 2000 .

[102]  τ-function for analytic curves , 2000 .

[103]  P. Deift Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[104]  Spohn,et al.  Universal distributions for growth processes in 1+1 dimensions and random matrices , 2000, Physical review letters.

[105]  Integrable structure of interface dynamics , 2000, Physical review letters.

[106]  Limiting Distributions for a Polynuclear Growth Model with External Sources , 2000, math/0003130.

[107]  H. Spohn,et al.  Scale Invariance of the PNG Droplet and the Airy Process , 2001, math/0105240.

[108]  A concise expression for the ODE's of orthogonal polynomials , 2001, math-ph/0109018.

[109]  Pavel Bleher,et al.  Random Matrix Models and Their Applications , 2001 .

[110]  Exact solution of the three-color problem on a random lattice , 2000, hep-th/0005190.

[111]  C. Vafa,et al.  Matrix models, topological strings, and supersymmetric gauge theories , 2002, hep-th/0206255.

[112]  A. Zee,et al.  RNA folding and large N matrix theory , 2001, cond-mat/0106359.

[113]  A Classification of Non-Hermitian Random Matrices , 2001, cond-mat/0110649.

[114]  Pierre Van Moerbeke Random matrices and permutations, matrix integrals and integrable systems , 2002 .

[115]  M. Praehofer,et al.  Stochastic Surface Growth , 2003 .

[116]  Kurt Johansson Discrete Polynuclear Growth and Determinantal Processes , 2003 .

[117]  Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem , 2002, nlin/0208002.

[118]  Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues , 2004, math-ph/0402053.

[119]  Random matrix theory and symmetric spaces , 2003, cond-mat/0304363.

[120]  Fluctuations of the One-Dimensional Polynuclear Growth Model in Half-Space , 2003, cond-mat/0307011.

[121]  Fluctuations of an atomic ledge bordering a crystalline facet. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[122]  Piero Barone,et al.  On the distribution of poles of Padé approximants to the Z-transform of complex Gaussian white noise , 2005, J. Approx. Theory.

[123]  LETTER TO THE EDITOR: A determinantal formula for the GOE Tracy Widom distribution , 2005, math-ph/0505012.

[124]  B. Eynard,et al.  2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals , 2005, hep-th/0502041.

[125]  Bertrand Eynard,et al.  Universal distribution of random matrix eigenvalues near the ‘birth of a cut’ transition , 2006, math-ph/0605064.

[126]  Mixed correlation function and spectral curve for the 2-matrix model , 2006, math-ph/0605010.

[127]  Bertrand Eynard,et al.  Formal Matrix Integrals and Combinatorics of Maps , 2006, math-ph/0611087.

[128]  S. Y. Lee,et al.  First Colonization of a Spectral Outpost in Random Matrix Theory , 2007, 0711.3625.

[129]  T. Sasamoto,et al.  Dynamical properties of a tagged particle in the totally asymmetric simple exclusion process with the step-type initial condition , 2007 .

[130]  B. Eynard,et al.  Invariants of algebraic curves and topological expansion , 2007, math-ph/0702045.

[131]  Fluctuation Properties of the TASEP with Periodic Initial Configuration , 2006, math-ph/0608056.

[132]  Marco Bertola,et al.  Boutroux curves with external field: equilibrium measures without a minimization problem , 2007, 0705.3062.

[133]  Z. Bai,et al.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES, A REVIEW , 2008 .

[134]  Harish-Chandra Integrals as Nilpotent Integrals , 2008, 0801.3452.

[135]  Alexei Borodin,et al.  Large Time Asymptotics of Growth Models on Space-like Paths II: PNG and Parallel TASEP , 2007, 0707.4207.

[136]  B. Eynard,et al.  Topological expansion of the chain of matrices , 2008, 0805.1368.

[137]  B. Eynard A Matrix model for plane partitions and TASEP , 2009 .

[138]  Celia Arnaud UP IN THE CLOUDS , 2009 .

[139]  Bertrand Eynard,et al.  A matrix model for plane partitions , 2009, 0905.0535.

[140]  B. Eynard,et al.  Some properties of angular integrals , 2008, 0805.4482.

[141]  V. Bouchard,et al.  Remodeling the B-Model , 2007, 0709.1453.

[142]  Patrick Desrosiers,et al.  Duality in random matrix ensembles for all β , 2008, 0801.3438.

[143]  P. Forrester Log-Gases and Random Matrices , 2010 .

[144]  T. Tao,et al.  Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge , 2009, 0908.1982.

[145]  John Harnad,et al.  Random Matrices, Random Processes and Integrable Systems , 2011 .

[146]  J. Baik,et al.  The Oxford Handbook of Random Matrix Theory , 2011 .

[147]  P. Ferrari,et al.  Finite Time Corrections in KPZ Growth Models , 2011, 1104.2129.

[148]  B. Eynard,et al.  Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture , 2012, 1205.1103.

[149]  B. Eynard,et al.  Torus Knots and Mirror Symmetry , 2011, 1105.2012.

[150]  J. Quastel,et al.  Continuum Statistics of the Airy2 Process , 2011, 1106.2717.

[151]  A. Guionnet,et al.  Asymptotic expansion of β matrix models in the multi-cut regime , 2013 .