# Random matrix theory

Yan Fyodorov (2011), Scholarpedia, 6(3):9886. | doi:10.4249/scholarpedia.9886 | revision #90306 [link to/cite this article] |

**Random Matrix Theory** (frequently abbreviated as RMT) is an active research area of modern Mathematics with input from Mathematical and Theoretical Physics, Mathematical Analysis and Probability, and with numerous applications, most importantly
in Theoretical Physics, Number Theory, and Combinatorics, and further in Statistics, Financial Mathematics, Biology
and Engineering & Telecommunications.

## Contents |

## Introduction

The main goal of the Random Matrix Theory is to provide understanding of the diverse properties (most notably, statistics of matrix eigenvalues) of matrices with entries drawn randomly from various probability distributions traditionally referred to as the random matrix ensembles. Three classical random matrix ensembles are the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE). They are composed respectively of real symmetric, complex Hermitian and complex self-adjoint quaternion matrices with independent, normally distributed mean-zero entries whose variances are adjusted to ensure the invariance of their joint probability density with respect to Orthogonal (respectively, Unitary or Symplectic) similarity transformations. Such invariance is also shared by the corresponding Lebesgue measures. If one keeps the requirement of invariance of the joint probability density of all entries but relaxes the property of entries being independent, one arrives at a broader classes of Invariant non-Gaussian Ensembles (Orthogonal, Unitary, or Symplectic). Note that being self-adjoint the matrices of those types have all their eigenvalues real.

Equally important are the Circular Ensembles composed of complex unitary matrices sharing the same invariance
properties of the measure as their Gaussian counterparts, but with eigenvalues confined to the unit circle
in the complex plane rather than to the real line.
Those ensembles are known as the Circular Orthogonal Ensemble (COE), the Circular Unitary Ensemble (CUE) and the Circular Symplectic
Ensemble (CSE). Finally deserves mentioning the so-called Ginibre Ensemble of matrices with independent, identically and normally distributed real, complex, or quaternion real entries, and no further constraints imposed. The corresponding eigenvalues are scattered in the complex plane.

## Brief History and Background Information

Although origins of RMT could be traced back to works by Wishart (1928) [1] and James (1954-1964) [2] in the field of Statistics (see also Hua's works in multivariate harmonic analysis [3]) the real start of the field is usually attributed to highly influential papers by Eugene Wigner in 1950's [4] motivated by applications in Nuclear Physics. Wigner suggested that fluctuations in positions of compound nuclei resonances can be described in terms of statistical properties of eigenvalues of very large real symmetric matrices with independent, identically distributed entries. The rational behind such a proposal was the idea that in the situation when it is hardly possible to understand in detail individual spectra associated with any given nucleus composed of many strongly interacting quantum particles, it may be reasonable to look at the corresponding systems as "black boxes" and adopt a kind of statistical description, not unlike thermodynamics approach to classical matter. A system in Quantum Mechanics can be characterized by a self-adjoint linear operator in Hilbert space, its Hamiltonian, which we may think informally of as a matrix of infinitely many dimensions. This suggests that input to such theory should be very general properties of the underlying generic Hamiltonians, most importantly such global features as the Hermiticity, the time-inversion invariance as well as other symmetries Hamiltonians may obey from general principles. Wigner hoped that the output of the model will be universal, that is independent on the detail and common to majority of systems sharing the corresponding symmetries. Along those lines Wigner succeeded in calculating the simplest nontrivial spectral characteristics of random real symmetric matrices with independent, identically distributed entries - the mean density of eigenvalues, and demonstrated that in the limit of large matrix size it is given quite generally by the so called Semicircular Law. Wigner also provided insights into statistics of separations between the neighbouring eigenvalues of such matrices (Wigner surmise).

Boosted by those works RMT problems attracted considerable attention and within the next few years many essential tools helping to analyse properties of random matrices were developed, most notably the method of orthogonal polynomials by Mehta and Gaudin [5]. Wigner's ideas were further substantiated by the seminal Dyson works [6] who gave important symmetry classification of Hamiltonians implying the existence of three major symmetry classes of random matrices - Orthogonal, Unitary and Symplectic, which cover the most relevant classical ensembles. Dyson also introduced the abovementioned circular versions of random matrix ensembles, developed a detailed theory of their spectra, and suggested a model of Brownian motion in random matrices ensembles which proved to be conceptually important and established a link to exactly soluble systems, such as the Calogero-Sutherland-Moser model.

Starting from the 1980's and further into 1990's the interest in RMT ideas, methods and results became widespread in the Theoretical Physics community. Motivated initially mainly by applications in Nuclear Physics (see [7] for an early influential review) RMT was further influenced by advances in the field of Quantum Chaos, such as the conjecture of Bohigas, Giannoni, and Schmit (BGS) [8] claiming statistical similarity between RMT spectra and highly excited energy levels of generic quantum systems whose classical counterparts show chaotic dynamical behaviour (see [9] for recent works towards justification of the BGS conjecture). From a different angle, somewhat similar ideas were substantiated and quantified by applications of powerful methods coming from the field of disordered systems and the Anderson localization, most importantly by supermatrix approach due to Efetov [10] and its further adaptation to RMT by Weidenmueller, Verbaarschot, and Zirnbauer [11] ( see [12] for a few recent reviews). Also influential were a highly successful maximal-entropy based RMT approach to statistics of electronic transport in quantum-coherent (mesoscopic) samples introduced by Mello and collaborators [13], as well as the idea of parametric dependence of spectra of disordered and chaotic systems, and their RMT counterparts, developed by Simons and Altshuler [14] (see also related works [15] on statistics of level curvatures, as well as early papers [16] on ensembles interpolating between GOE and GUE and [17] on an impressively accurate verification of many aspects of RMT in acoustic waves experiments). Those and related developments resulted in a considerable broadening of the list of random matrix ensembles relevant to physical applications and amenable to analytical treatment beyond the classical Wigner-Dyson list. In particular, ensembles of random matrices which are not invariant with respect to changes of the basis attracted considerable attention, e.g. Band Random Matrices [18], which allowed for the effects of the Anderson localization to be correctly incorporated at the level of RMT. Moreover, the Dyson three-fold symmetry classification of underlying Hamiltonians was discovered to be insufficient for describing spectra appearing in Quantum Chromodynamics [19] as well as in disordered superconducting structures and was replaced by a more comprehensive list [20]. Around the same time systematic investigations of non-Hermitian deformations of random matrix ensembles were undertaken due to their relevance for description of statistics of S-matrix poles (resonances) in quantum chaotic scattering [21] and in Quantum Chromodynamics with non-zero chemical potential [22], which also lead to a general boost of interest in their properties [23].

On the other hand, from a very different perspective deep connections to the Random Matrix Theory were discovered in the models of 2-dimensional Quantum Gravity [24]. That line of research could be traced back to the t'Hooft's idea that in field theories with a large gauge group (as e.g. U(N)) the diagrammatic expansion will be dominated in the large-N limit by planar Feynman diagrams. In the RMT context such an expansion was first introduced in the influential paper by Brézin, Itzykson, Parisi and Zuber [25]. Further developments showed that a method particularly suited to calculate the genus (or topological) expansion in 1/N turns out to be the loop equation method, see [26] and references therein. The ensuing links of random matrices to the theory of integrable systems, in particular, to Painleve transcendents and Toda/KdV hierarchies [27] and to diverse combinatorial problems [28] became an area of thriving activity.

Until the end of 1980's the research on properties of random matrices in Mathematics community was seemingly not as nearly as intensive as in Theoretical Physics. Among a few early mathematically rigorous and influential papers one could mention the works by Marcenko and Pastur describing the spectrum of large random covariance matrices [29], and by Arnold, Pastur, and Furedi & Komlos on various aspects of the eigenvalue distributions in ensembles of random matrices with independent entries [30]. An important link of RMT to profound number-theoretical problems was provided by Montgomery [31] who revealed a certain similarity between eigenvalues of GUE/CUE matrices to those of zeroes of the Riemann zeta-function (Montgomery conjecture). On the other hand, needs of numerical analysis stimulated interests in the condition numbers of random matrices [32].

The situation changed considerably in 1990's after a few independent developments in various branches of Mathematics (among others, development of the "free probability" theory by D. Voiculescu and its relations to random matrices of infinite size [33]) lead to an essential boost of interest in various aspects of the Random Matrix Theory. First of all, the wealth of information obtained by heuristic methods of theoretical physics called for a proper understanding by mathematical standards. To that end, considerable efforts were directed towards proving conjectured universality [34] of correlations between eigenvalues of random matrices beyond classical (Gaussian or Circular) Wigner-Dyson ensembles, that is independence of their properties of the choice of the distribution of matrix elements for general RMT ensembles in the limit of large dimension. First, such a universality was verified for a rather general class of invariant ensembles [35], the line of research resulting, in particular, in development of the powerful Riemann-Hilbert approach to the asymptotics of orthogonal polynomials [36]. More recently similar universality was extended also to a broad class of ensembles with independent, identically distributed entries [37]. Deserve mentioning also works [38] on the large deviation statistics of spectral measure and on distribution of the number of eigenvalues in a given interval of spectrum, as well as the paper [39] which nontrivially extends the Dyson work on Brownian motion of eigenvalues by relating it to non-intersecting random walks.

Around the same time the statistics of the largest eigenvalue in Gaussian Ensembles attracted considerable attention [40]. Its probability density first derived by C. Tracy and H. Widom [41] appeared to be highly universal [42] and emerged in such important combinatorial problems as the distribution of the length of the longest increasing subsequence of random permutations [43] as well as in applications in statistical mechanics: statistics of the height of random surfaces obtained by polynuclear growth [44] and the lowest energy state as well as the free energy of a directed polymer in random environment [45]. Yet another highly influential development followed the works by Keating and Snaith [46] in providing powerful evidences in favour of a very intimate connection between the properties of characteristic polynomials of random matrices and the moments of the Riemann zeta-function (and other L-functions) along the so-called critical line. That line of research stimulated interest in general correlation properties of the characteristic polynomials of RMT ensembles [47]. Ideas coming from the field of numerical matrix analysis lead to a fruitful and promising enrichment of the Random Matrix theory by "beta-ensembles" due to Dumitriu and Edelman [48] with a continuous positive parameter beta characterizing the statistics of their eigenvalues, the three discrete values beta=1,2 and 4 corresponding to the classical Orthogonal, Unitary, and Symplectic RMT ensembles. Apart from that, a steady growth of attention to various aspects of random matrix ensembles of non-Hermitian matrices with eigenvalues scattered in the complex plane [49] is to be mentioned, extending and generalizing early works by Ginibre [50] and Girko [51]. Among other actively researched topics deserve mentioning works on singular values distributions and eigenvalues of random covariance matrices [52], important, in particular, for applications in quantum information context [53] and for the analysis of multivariate data in time series appearing in financial mathematics [54]. Actively investigated were also random matrix ensembles with multifractal eigenvectors and/or the so-called critical eigenvalue statistics [55], ensembles of heavy-tailed random matrices [56], of sparse random matrices [57], and of Euclidean random matrices [58], as well as random matrices in external source and coupled in a chain [59]. Increasingly important role in many RMT developments continue to play Harish-Chandra-Itzykson-Zuber integration formula [60] and its extensions, as well as the Selberg Integral, see [61] for a recent review.

## References

- Wishart, J. Generalized product moment distribution in samples. Biometrika 20 A (1928) no. 1/2, 32-52 [Correction: ibid. 20 A.(1928) p.425]
- James, A. T. Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. v. 25 (1954) no.1, 40-75; James, A. T. The Distribution of the Latent Roots of the Covariance Matrix. Ann. Math. Statist. v. 31 (1960) no.1, 151-158; James, A. T. Distributions of Matrix Variates and Latent Roots Derived from Normal Samples. Ann. Math. Statist. v. 35 (1964) no.2, 475-501
- Hua, Lo-keng Harmonic analysis of functions of several complex variables in the classical domains. American Mathematical Society, Rhode Island (1963) [Chinese edition: (1958)].
- Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. v.62 (1955) no.3, 548-564; Wigner, E. Characteristic vectors of bordered matrices of infinite dimensions II. Ann. of Math. v.65 (1957) no.2, 203-207; Wigner, E. On the distribution of the roots of certain symmetric matrices. Ann. of Math. v.67 (1958) no.2, 325-326; Wigner, E. Random Matrices in Physics. SIAM Reviews v.9 (1967) No. 1, 1-23
- Mehta, M. L. On the statistical properties of the level-spacings in nuclear spectra. Nuclear Phys. v. 18 (1960) 395-419 ; Mehta, M. L.; Gaudin, M. On the density of eigenvalues of a random matrix. Nuclear Phys. 18 (1960) 420-427; Gaudin, M. Sur la loi limite de l'espacement des valeurs propres d'une marice aleatoire. Nuclear Phys. v. 25 (1961) 447-458
- Dyson, F. J. Statistical Theory of the energy levels of complex systems, I, II, & III, J. Math. Phys. v.3 (1962) no.1, 140-175 ; Dyson, F. J. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. v. 3 (1962) no.6, 1191-1198. ; Dyson, F J. The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. v.3 (1962) no.6, 1199-1215. ; Dyson, F. J.; Mehta, M. L. Statistical theory of the energy levels of complex systems. IV. J. Math. Phys. v.4 (1963) no.5, 701-712; Mehta, M. L.; Dyson, F. J. Statistical theory of the energy levels of complex systems. V. J. Math. Phys. v.4 (1963) no.5, 713-719 ; Dyson, F. J. Correlations between eigenvalues of a random matrix. Commun. Math. Phys. v. 19 (1970) no. 3, 235-250; Dyson, F. J. Fredholm determinants and inverse scattering problems. Commun. Math. Phys. v. 47 (1976) no. 2, 171-183
- Flores, J; French, J.B.; Mello, P.A., Pandey, A; Wong S.S.M. Random Matrix Physics: Spectrum and Strength Fluctuations. Rev. Mod. Phys. v. 53 (1981) no.3, 385-479
- Bohigas, O.; Giannoni, M.-J.; Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. v. 52 (1984), no. 1, 1-4.
- Muzykantskii, B.A; Khmelnitskii, D.E. Effective action in the theory of quasi-ballistic disordered conductors. JETP Lett. v. 62 (1995) no.1, 76-82; Andreev, AV; Agam, O; Simons, B.D.; Altshuler, B.L. Quantum chaos, irreversible classical dynamics, and random matrix theory. v. 76 (1996) no. 21 , 3947-3950 ; Mueller, S. ; Heusler, S.; Braun P.; Haake F.; and Altland A. Semiclassical Foundation of Universality in Quantum Chaos. Phys. Rev. Lett. v. 93 (2004), no. 1, 014103 ; Heusler, S.; Mueller, S.; Altland, A. ; Braun, P.; Haake, F. Periodic-Orbit Theory of Level Correlations. Phys. Rev. Lett. v. 98 (2007), no. 4, 044103
- Efetov, K. B. Supesymmetry and theory of disordered metals. Adv. Phys. v. 32 (1983) no. 1, 53-127
- Verbaarschot, J. J. M.; Weidenmüller, H. A.; Zirnbauer, M.R. Grassmann integration in stochastic quantum physics: the case of compound-nucleus scattering. Phys. Rep. v. 129 (1985), no. 6, 367-438.
- Zirnbauer, M.R. The supersymmetry method of random matrix theory. [arXiv:math-ph/0404057]; Guhr, T. Supersymmetry in Random Matrix Theory. [arXiv:1005.0979]
- Mello, P.A.; Pereyra, P.; Seligman, T.H. Information theory and statistical nuclear reactions. I. General theory and applications to few-channel problems. Ann. Phys. v. 161 (1985) no. 2 , 254-275 ; Mello, P.A.; Stone, A.D. Maximum-Entropy Model for Quantum-Mechanical Interference Effects in Metallic Conductors. Phys. Rev. B v. 44 (1991) no. 8, 3559-3576 ; Baranger, H.U. ; Mello, P.A. Mesoscopic Transport Through Chaotic Cavities - a Random S-matrix Theory Approach. Phys. Rev. Lett. v. 73 (1994) no. 1, 142-145;
- Simons, B.D. ; Altshuler, B.L. Universalities in the spectra of disordered and chaotic systems. Phys. Rev. B v.48 (1993) no. 8, 5422-5438; Simons, B.D. ; Lee, P.A. and Altshuler, B.L. Matrix Models, one-dimensional fermions and Quantum Chaos. Phys. Rev. Lett. v. 72 (1994) no. 1, 64-67
- Gaspard, P.; Rice, S.A., Miekeska. H.-J., Nakamura K. Parametric motion of energy levels - curvature distribution. Phys. Rev. A v. 42 (1990) no.7, 4015-4027; Zakrzewski. J.; Delande, D. Parametric motion of energy levels in chaotic-quantum systems 1. Curvature distribution. Phys. Rev. E. v. 47 (1993) no. 3, 1650-1664; von Oppen, F. Distribution of eigenvalue curvatures of chaotic-quantum systems. Phys. Rev. Lett. v. 73 (1994) no. 6, 798-801; von Oppen, F. Exact Distributions of Eigenvalue Curvatures for Time-Reversal-Invariant Chaotic Systems. Phys. Rev. E v. 51 (1995) no. 3, 2647-2650; Fyodorov, Y.V.; Sommers, H.J. Universality of "level curvature" distribution for large random matrices: systematic analytical approaches. Z. Phys. B v.99 (1995) no. 1, 123-135
- Pandey, A; Mehta, M.L. Gaussian Ensembles of Random Hermitian Matrices intermediate between Orthogonal and Unitary ones. Commun. Math. Phys. v. 87 (1983), no. 4, 449-468
- Ellegaard, C. ; Guhr, T. ; Lindemann, K. ; Lorensen, H.Q. ; Nygard J. ; Oxborrow, M. Spectral Statistics of Acoustic Resonances in Aluminum Blocks. Phys. Rev. Lett. v. 75 (1995) no.8, 1546-1549; Ellegaard,C. ; Guhr, T. ; Lindemann, K. ; Lorensen, H.Q. ; Nygard J. ; Oxborrow, M. J. Symmetry Breaking and Spectral Statistics of Acoustic Resonances in Quartz Blocks. Phys. Rev. Lett. v. 77 (1996) no. 24, 4918-4921
- Casati, G.; Molinari, L.; Izrailev, F. Scaling Properties of Band Random Matrices. Phys. Rev. Lett. v.64 (1990), no. 16 ,1851-1854; Fyodorov, Y. V.; Mirlin, A. D. Scaling properties of localization in random band matrices: a $\sigma$-model approach. Phys. Rev. Lett. v.67 (1991), no. 18, 2405-2409. Fyodorov, Y. V.; Mirlin, A. D. Statistical properties of eigenfunctions of random quasi 1d one-particle Hamiltonians. Int. J. Mod. Phys. B v. 8 (1994) no. 27, 3795-3842;
- Shuryak, E. V. ; Verbaarschot, J.J.M. Random Matrix Theory And Spectral Sum Rules For The Dirac Operator In QCD Nucl. Phys. A v. 560 (1993) no.1, 306-320 [arXiv:hep-th/9212088]; Verbaarschot, J.J.M. ; Zahed, I. Spectral Density of the QCD Dirac Operator Near Zero Virtuality. Phys. Rev. Lett. v. 70 (1993) no.25, 3852-3855; Verbaarschot, J.J.M. The Spectrum of the QCD Dirac operator and chiral random matrix theory: The Threefold way. Phys. Rev. Lett. v. 72 (1994) no. 16, 2531-2533 [arXiv:hep-th/9401059]; Verbaarschot, JJM ; Wettig, T. Random matrix theory and chiral symmetry in QCD. Ann. Rev. Nucl. Part. Sci. v. 50 (2000) no. 343-410
- Zirnbauer, M R. Riemannian symmetric superspaces and their origin in random-matrix theory. J. Math. Phys. v. 37 (1996), no. 10, 4986-5018. ; Altland, A; Zirnbauer, M R. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B v.55 (1997) 1142–1161
- Sokolov, V.V.; Zelevinsky, V.G. Dynamics and Statistics of Unstable Quantum States. Nucl. Phys. A. v. 504 (1989), no.3, 562-588; Haake, F.; Izrailev F.; Lehmann, N.; Saher, D.; Sommers H.J. Statistics of Complex levels of Random Matrices for Decaying Systems. Z. Phys. B: Cond. Matt. v. 88 (1992) no.3, 359-370; Fyodorov YV; Sommers H.-J. Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance. J. Math. Phys. v.38 (1997) v.4, 1918-1981; Sommers, H.J.; Fyodorov, Y.V.; Titov, M. S-matrix poles for chaotic quantum systems as eigenvalues of complex symmetric random matrices: from isolated to overlapping resonances. J. Phys. A: Math. Gen. v.32 (1999) no. 5, L77-L85; Fyodorov, Y.V.; Khoruzhenko, B.A. Systematic analytical approach to correlation functions of resonances in quantum chaotic scattering. Phys. Rev. Lett. v. 83 (1999) no. 1, 65-68
- Stephanov, M.A. Random Matrix Model of QCD at Finite Density and the Nature of the Quenched Limit. Phys. Rev. Lett. 76 (1996), no. 24, 4472–4475; Halasz, M.A.; Osborn, J.C.; Verbaarschot, J.J.M. Random matrix triality at nonzero chemical potential. Phys. Rev. D 56 (1997) 7059–7062; Akemann, G. Microscopic Correlation Functions for the QCD Dirac Operator with Chemical Potential. Phys. Rev. Lett. 89 (2002) 072002; Osborn, J.C., "Universal results from an alternate random matrix model for QCD with a baryon chemical potential," Phys. Rev. Lett. v. 93 (2004) no.22, 222001 [arXiv:hep-th/0403131];
- Sommers, H.J. ; Crisanti, A. ; Sompolinsky, H ; Stein, Y. Spectrum of Large Random Asymmetric Matrices. Phys. Rev. Lett. v.60 (1988) no. 19, 1895-1898; Bai, Z.D. Circular Law. Ann. Prob. v. 25 (1997), no. 1, 494-529; Fyodorov, Y.V.; Khoruzhenko, B. A.; Sommers, H.-J. Almost Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre Eigenvalue Statistics. Phys. Rev. Lett. 79, (1997) 557–560; Efetov, K.B. Directed quantum chaos. Phys. Rev. Lett. v. 79 (1997), no. 3, 491-494; Edelman, A. The probability that a random real gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivar. Anal. v. 60 (1997), no. 2, 203-232; Feinberg, J. ; Zee, A. Non-hermitian random matrix theory: Method of hermitian reduction. Nucl. Phys. B v. 504, no. 3 ( 1997) 579-608; Janik, R. A.; Nowak, M. A.; Papp, G.; Zahed, I. Non-hermitian random matrix models. Nuclear Physics B v. 501, no. 3 (1997)pp 603-642; Goldsheid, I.Ya.; Khoruzhenko, B.A. Distribution of Eigenvalues in Non-Hermitian Anderson Models. Phys. Rev. Lett. 80 (1998) 2897–2900;
- Brézin, É.; Kazakov, V. A. Exactly solvable field theories of closed strings. Phys. Lett. B v. 236 (1990) no. 2, 144-150. Gross, D. J.; Migdal, A. A. Nonperturbative two-dimensional quantum gravity. Phys. Rev. Lett. 64 (1990), no. 2, 127-130. Douglas, M. R.; Shenker, S. H. Strings in less than one dimension. Nuclear Phys. B v. 335 (1990), no. 3, 635-654. ; Kontsevich, M. Commun. Math. Phys. v. 147 (1992), no. 1 , 1-23
- Brézin, É.; Itzykson, I.; Parisi, G.; and Zuber, J.-B. Planar Diagrams. Comm. Math. Phys. v. 59 (1978) no. 1, 35-51.
- Ambjorn, J. ; Chekhov, L; Kristjansen C.F; and Makeenko Yu., Matrix model calculations beyond the spherical limit, Nucl. Phys. B v. 404 no. 1-2(1993) 127-172 [Erratum-ibid. v. 449 (1995) no. 3, 681-681] [arXiv:hep-th/9302014]; Eynard, B. Topological expansion for the 1-hermitian matrix model correlation functions. JHEP v.11 (2004), 031; Eynard B.; Orantin N. Invariants of algebraic curves and topological expansion. Comm. Numb. Th. Phys. v.1 (2007), no. 2, 347-452 [arXiv: math-ph/0702045]
- Gerasimov, A.; Marshakov A.; Mironov A.; Morozov A.; Orlov A. Matrix Models of 2-Dimensional Gravity and Toda Theory. Nucl. Phys. B v. 357 (1991), no: 2-3, 565-618 ; Tracy, C.A. ; Widom, H. Fredholm Determinants, Differential Equations, and Matrix Models. Commun. Math. Phys. v.163 (1994) no. 1, 33-72 ; Adler, M.; Shiota, T.; van Moerbeke, P., Random matrices, vertex operators and the Virasoro algebra. Phys. Lett. A v. 208 (1995) no. 1-2, 67-78 ; Adler, M.; van Moerbeke, P. Hermitian, symmetric and simplectic random matrix ensembles: PDEs for the distribution of the spectrum. Ann. Math. v. 153 (2001) 149-189; Forrester, P.J.; Witte, N.S. Application of the tau-function theory of Painleve equations to random matrices: PIV, PII and the GUE. Comun. Math. Phys. v. 219 (2001), no. 2, 357-398; Marshakov, A.; Wiegmann, P.; Zabrodin, A. Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. v. 227 (2002) no. 1, 131-153; Kanzieper, E. Replica field theories, Painleve transcendents, and exact correlation functions. Phys. Rev. Lett. v. 89 (2002) no.25, 250201
- Di Francesco, P. 2D quantum gravity, matrix models and graph combinatorics. in: Brezin, E. et al (eds.) Applications of random matrices in physics, 33--88, NATO Sci. Ser. II Math. Phys. Chem., 221, Springer, Dordrecht, 2006.
- Marcenko, V. A.; Pastur, L. A. Distribution of eigenvalues in certain sets of random matrices. (Russian) Mat. Sb. (N.S.) 72 (114) 1967 507--536
- Arnold, L. On Asymptotic Disribution of Eigenvalues of Random Matrices. J. Math. Anal. Appl. v. 20 (1967) no. 2, 262-?.; Pastur L.A. On the spectrum of random matrices. Theor. and Math. 10 (1972), No. 1, 67-74; Furedi, Z.; Komlos, J. The Eigenvalues of Random Symmetric Matrices. Combinatorica. v. 1 (1981) no. 3, 233-241
- Montgomery, H. L. The pair correlation of zeros of the zeta function. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), 181-193. Amer. Math. Soc., Providence, R.I., 1973.
- Edelman, A. Eigenvalues and Condition numbers of Random Matrices. SIAM J. Matr. Anal. Appl. v.9 (1988), no. 4, 543-560;
- Voiculescu, D. V. Limit Laws for Random Matrices and Free Products. Invent. Math. v. 104 (1991) no.1, 201-220
- Brézin, É. , Zee A. Universality of the Correlation Function between Eigenvalues of Large Random Matrices. Nucl. Phys. B. v. 402 (1993) no. 3, 613-627
- Pastur, L.; Shcherbina, M. Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. v. 86 (1997), no. 1-2, 109-147; Zinn-Justin, P. Universality of correlation functions of hermitian random matrices in an external field. Comm. Math. Phys. v. 194 (1998), no. 3, 631-650 ; Kuijlaars, A.B.J. ; Vanlessen, M. Universality for eigenvalue correlations from the modified Jacobi unitary ensemble. Int. Math. Res. Notices 30 (2002), 1575-1600; Kuijlaars, A.B.J. ; Vanlessen, M. Universality for eigenvalue correlations at the origin of the spectrum. Comm. Math. Phys. 243 (2003) No. 1, 163-191;
- Bleher, P.; Its, A. Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1999), no. 1, 185-266. Deift, P.; Kriecherbauer, T.; McLaughlin, K. T.-R.; Venakides, S.; Zhou, X. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. v. 52 (1999), no. 11, 1335-1425.
- Johansson, K. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. v. 215 (2001), no.3, 683-705. ; Erdos, L.; Peche, S., Ramirez, J; Schlein B.; Yau, H-T. Bulk universality for Wigner matrices. Comm. Pure Appl. Math. v. 63 (2010), no.7, 895-925; Tao T. and Vu. V. Random matrices: Universality of local eigenvalue statistics. Acta Math. v.206 (2011), 127-204.
- Ben Arous, G.; Guionnet, A. Large deviations for Wigner's law and Voiculescu's non-commutative entropy. Prob. Theor. Rel. Fields v. 108 (1997) no. 4, 517-542; Costin, O. ; Lebowitz, J.L. Phys. Rev. Lett. v. 75 (1995) no. 1 , 69-72
- Grabiner, D.J. Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincare-Prob. Stat. v. 35 (1999) no. 2, 177-204
- Forrester, P.J. The spectrum edge of random matrix ensembles. Nucl. Phys. B. v. 402 (1993), no. 3, 709-728; Nagao, T; Forrester, P.J. Asymptotic Correlations at the spectrum edge of random matrices.
- Tracy, C. A.; Widom, H. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. v. 159 (1994), no. 1, 151-174.
- Soshnikov, A. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. v. 207 (1999), no. 3, 697-733. ; Johnstone, M. On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. v. 29 (2001) No. 2, 295-327; Deift P., Gioev, D. Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Comm. Pure Appl. Math. v. 60 (2007) no. 6 , 867-910; Tao, T and Vu, V. Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge. Commun. Math. Phys. v.298 (2010) no. 2, 549-572
- Baik, J; Deift, P.; Johansson, K. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. v. 12 (1999), no. 4, 1119-1178. Okounkov, A. Random matrices and random permutations. Internat. Math. Res. Notices 2000, no. 20, 1043--1095.; Borodin, A.; Okounkov, A.; Olshanski, G. Asymptotics of Plancherel Measures for Symmetric Groups. J. Am. Math. Soc. v. 13 (2000) no. 3, 481-515
- Prähofer, M.; Spohn, H. Scale invariance of the PNG droplet and the Airy process. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J. Statist. Phys. 108 (2002), no. 5-6, 1071-1106.
- Johansson, K. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437--476. Sasamoto T., Spohn H. Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Physics B v. 834, no. 3, (2010) pp 523-542 ; Amir, G.; Corwin, I.; Quastel, J. Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+ 1 dimensions. Communications on Pure and Applied Mathematics, n/a. doi: 10.1002/cpa.20347 ; Calabrese, P.; Le Doussal, P., Rosso, A. Free-energy distribution of the directed polymer at high temperature. EPL 90 (2010) 20002 ; Dotsenko, V. Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers. EPL 90 (2010) 20003
- Keating, J. P.; Snaith, N. C. Random matrix theory and $\zeta(1/2+it)$. Comm. Math. Phys. v. 214 (2000), no. 1, 57-89. Keating, J. P.; Snaith, N. C. Random matrix theory and $L$-functions at $s=1/2$. Comm. Math. Phys. 214 (2000), no. 1, 91-110.
- Brézin, É. and Hikami S. Characteristic Polynomials of Random Matrices. Comm. Math. Phys. v 214 (2000) No. 1, 111-135; Hughes, C.P.; Keating, J.P.; O'Connell, N. On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. v. 220 (2001) no. 2, 429-451; Conrey, J.B.; Farmer, D.W.; Keating, J.P.; Rubinstein, M.O. ; Snaith, N.C. Autocorrelations of Random Matrix polynomials. Comm. Math. Phys. v. 237 (2003) no. 3, 365-395; Strahov, E. ; Fyodorov, Y.V. Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach. Comm. Math. Phys. v. 241 (2003) no. 2-3, 343-382; Borodin, A. ; Strahov, E. Averages of characteristic polynomials in random matrix theory. Comm. Pure Appl. Math. v. 59 (2006) no. 2, 161-253; Conrey, B. ; Farmer, D. W. ; Zirnbauer, M. R. Autocorrelation of ratios of L-functions. Comm. Numb. Theor. Phys. v.2 (2008) no. 3, 593-636; Desrosiers, P. Duality in random matrix ensembles for all beta. Nucl. Phys. B v. 817 (2009) no. 3, 224-251; Osipov, Al. V. ; Kanzieper, E. Correlations of RMT characteristic polynomials and integrability: Hermitean matrices. Ann. Phys. v. 325 (2010) no. 10 , 2251-2306
- Dumitriu, I. ; Edelman, A. Matrix models for beta ensembles. J. Math. Phys. 43 (2002), no. 11, 5830-5847. Edelman, A. ; Rao, N. R. Random matrix theory. Acta Numer. 14 (2005), 233-297.
- Zyczkowski, K; Sommers, H.-J. Truncations of random unitary matrices. J. Phys. A: Math. Gen. v. 33 (2000) no.10, 2045--2057; Fyodorov, Y. V.; Sommers, H.-J. Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A: Math.Gen. v. 36 (2003), no. 12, 3303-3347. ; Wiegmann P. and Zabrodin, A. Large scale correlations in normal non-Hermitian matrix ensembles. J. Phys. A: Math. Gen. v. 36 (2003) no.12, 3411-3424 ; Splittorff K. and J.~J.~M.~Verbaarschot J.J.M., Factorization of correlation functions and the replica limit of the Toda lattice equation, Nucl. Phys. B v. 683 (2004) no.3, 467-507 [arXiv:hep-th/0310271]; Akemann, G.; Kanzieper, E. Integrable structure of Ginibre's ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. v.129 (2007), no. 5-6, 1159-1231; Tao, T; Vu, V. From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices. Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 3, 377-396. ; Guionnet, A. ; Krishnapur, M ; Zeitouni, O. The single ring theorem. Arxiv preprint arXiv:0909.2214;
- Ginibre, J. Statistical Ensembles of Complex, Quaternion, and Real Matrices. J. Math. Phys. v. 6,(1965) no.3, 440-450
- Girko, V. L. Circular Law. Theory Probab. Appl. v. 29 (1985) no. 4, 694-706
- Silverstein, J.W.; Bai, Z.D. On the empirical distribution of eigenvalues of a class of large dimensional random matrices. J. Multivar. Anal. v. 54 (1995) no. 2, 175-192; Sengupta, A.M.; Mitra, P.P. Distributions of singular values for some random matrices. Phys. Rev. E v. 60 (1999) no. 3, 3389-3392; Bai, Z.D., Silverstein, J.W. CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Prob. v. 32 (2004) no. 1A, 553-605; Tao, T; Vu, V. Random Matrices: The distribution of the smallest singular value. Geom. Funct. Anal. v. 20 (2010) no. 1, 260-297
- Zyczkowski, K.; Sommers, H. J. Induced measures in the space of mixed quantum states. J. Phys. A: Math. Gen. v. 34 (2001) no. 35, 7111-7125;
- Laloux, L.; Cizeau, P.; Bouchaud J.P.; Potters, M. Noise dressing of financial correlation matrices. Phys. Rev. Lett. v 83 (1999) no. 7 , 1467-1470 ; Plerou, V. ; Gopikrishnan, P., Rosenow, B.; Amaral, L.A.N.; Stanley H.E. Universal and nonuniversal properties of cross correlations in financial time series. Phys. Rev. Lett. v. 83 (1999) no. 7 , 1471-1474; Plerou, V.; Gopikrishnan P.; Rosenow B.; Amaral, L.A.N.; Guhr T.; and Stanley, H.E. A Random Matrix Approach to Cross--Correlations in Financial Data. Phys. Rev. E v.65 (2002), no.6, 066126
- Moshe M.; Neuberger, H; Shapiro, B.; Generalized ensemble of random matrices. Phys. Rev. Lett. v. 73 (1994) no. 11, 1497-1500; Mirlin, A.D; Fyodorov, Y.V ; Dittes, F.M.; Quezada, J. ; Seligman, T.H. Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices. Phys. Rev. E v. 54 (1996) 3221–3230; Kravtsov, V.E.; Muttalib, K.A. New class of random matrix ensembles with multifractal eigenvectors. Phys. Rev. Lett. v. 79 (1997) no. 10, 1913-1916; Bogomolny, E.B., Gerland, U.; Schmit, C. Models of intermediate spectral statistics. Phys.Rev. E v. 59 (1999) no.2, R1315-R1318; Mirlin, A.D. ; Evers. F. Multifractality and critical fluctuations at the Anderson transition. Phys. Rev. B 62, (2000) 7920–7933; Kravtsov, V.E.; Tsvelik, A.M. Energy level dynamics in systems with weakly multifractal eigenstates: Equivalence to one-dimensional correlated fermions at low temperatures. Phys. Rev. B v. 62 (2000) no. 15, 9888-9891; Fyodorov, Y.V.; Ossipov, A.; Rodriguez, A. The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices. J.Stat. Mech. (2009), L12001
- Cizeau, P.; Bouchaud, J.P. Theory of Levy Matrices. Phys. Rev. E v. 50 (1994) no. 3, 1810-1822; Burda, Z.; Janik, R.A.; Jurkiewicz, J.; Nowak, M.A.; Papp, G.; Zahed, I. Free random Levy matrices. Phys. Rev. E v. 65 (2002) no. 2, 021106; Soshnikov, A. Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails. Elec. Comm. Prob. v. 9 (2004), 82-91; Ben Arous, G.; Guionnet, A. The spectrum of heavy tailed random matrices. Comm. Math. Phys. v. 278 (2008) no. 3 , 715-751
- Rodgers, G.J., Bray, A.J. Density of States of Sparse Random Matrices. Phys. Rev. B v. 37 (1988) no. 7, 3557-3562 ; Mirlin, A.D.; Fyodorov, Y.V. Universality of Level Correlation Functions of Sparse Random Matrices. J. Phys. A: Math. Gen. v. 24 (1991) no. 10, 2273-2286; Fyodorov, Y.V.; Chubykalo, O.A.; Izrailev, F.M.; Casati, G. Wigner random banded matrices with sparse structure: Local spectral density of states. Phys. Rev. Lett. v. 76 (1996) no. 10, 1603-1606; Bauer, M., Golinelli, O. Random incidence matrices: Moments of the spectral density. J. Stat. Phys. v. 103 (2001), no. 1-2, 301-337 ; Semerjian, G.; Cugliandolo, L.F. Sparse random matrices: the eigenvalue spectrum revisited. J. Phys. A: Math. Gen. v. 35 (2002), no. 23, 4837-4851; Kuhn, R. Spectra of sparse random matrices. J. Phys. A: Math. Theor. v. 41 (2008), no. 29, 295002; Rogers, T. ; Castillo. I.P.; Kuhn, R., Takeda, K. Cavity approach to the spectral density of sparse symmetric random matrices. Phys. Rev. E. v. 78 (2008), no. 3, 031116
- Mezard, M.; Parisi, G.; Zee, A. Spectra of euclidean random matrices. Nucl.Phys. B. v. 559 (1999) no. 3, 689-701
- Zinn-Justin, P. Random hermitian matrices in an external field. Nucl. Phys. B. v. 497 (1997), no. 3, 725-732; Eynard, B.; Mehta, M.L. Matrices coupled in a chain: I. Eigenvalue correlations. J. Phys. A: Math. Gen. v. 31 (1998), no. 19, 4449-4456; Bleher, P.; Kuijlaars, A.B.J. Large n limit of Gaussian random matrices with external source, part I. Comm. Math. Phys. v. 252 (2004), v. 1-3, 43-76; Aptekarev, A.I., Bleher, P.M., Kuijlaars, A.B.J. Large n limit of Gaussian random matrices with external source, Part II. Comm. Math. Phys. v. 259 (2005), no. 2, 367-389
- Harish-Chandra. Differential operators on a semisimple Lie algebra. Amer. J. Math. v. 79 (1957), 87–120 ; Itzykson C., Zuber, J.B. Planar Approximation II. J. Math. Phys. v. 21 (1980), no. 3, 411-421;
- Forrester, P. J.; Warnaar, S. O. The importance of the Selberg integral. Bull. Amer. Math. Soc. (N.S.) v. 45 (2008), no. 4, 489–534.

## Recommended Reading

- Voiculescu, D. V. ; Dykema, K. J.; Nica, A. Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. CRM Monograph Series, 1. American Mathematical Society, Providence, RI, 1992. ISBN 0-8218-6999-X
- Di Francesco, P. ; Ginsparg, P ; Zinn-Justin, J. 2D gravity and random matrices. Phys. Rep. v. 254 (1995) no. 1-2, 1-133
- Beenakker C.W.J. Random-matrix theory of quantum transport. Rev. Mod. Phys. v.69 (1997) No. 3, 731-808
- Guhr, T.; Mueller-Groeling A.; Weidenmueller H.A. Random Matrix Theories in Quantum Physics: Common Concepts. Phys.Rep. v.299 (1998) pp 189-425
- Katz, N.M.; Sarnak, P. Random Matrices, Frobenius Eigenvalues, and Monodromy (Colloquium Publications . Amer Mathematical Soc. 1998), 419 pp
- Deift, P. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach (Courant Lecture Notes; Amer Mathematical Soc. 2000) 261 pp.
- Mehta, M.L. Random matrices, Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp
- Tulino, A. ; Verdu, S. Random Matrix Theory and Wireless Communications. Foundations and Trends in Communications and Information Theory. Now Publishers Inc. (2004) 192 pp
- Recent Perspectives in Random Matrix Theory and Number Theory (London Mathematical Society Lecture Note Series 322; ed. by Mezzardi F.; Snaith N. C.) Cambridge Univesity Press (2005), 518 pp
- Hiai F. and Petz D. The Semicircle Law, Free Random Variables and Entropy (Mathematical Surveys and Monographs. Amer Math. Soc. 2007) 376pp
- Anderson, G.W ; Guionnet, A; Zeitouni O. An Introduction to Random Matrices. Studies in Advanced Mathematics, v. 118, Cambridge University Press, 2009
- Deift. P; Gioev D. Random Matrix Theory: Invariant Ensembles and Universality. (Courant Lecture Notes; Amer Mathematical Soc. 2009) 217 pp.
- Bai, Z.D.; Silverstein J. W. Spectral analysis of Large Dimensional Random Matrices. (Springer Series in Statistics, 2nd ed., 2010), 552 pp
- Forrester, P.J. Log-Gases and Random Matrices (LMS-34) (London Mathematical Society Monographs), Princeton University Press, 2010, 808 pp
- Mitchell, G.E.; Richter, A., Weidenmuller, H.A. Random matrices and chaos in nuclear physics: Nuclear reactions Rev. Mod. Phys. v. 82 (2010), no. 2, 539-589
- Oxford Handbook on Random Matrix theory, edited by Akemann G.; Baik, J. ; Di Francesco P. , Oxford University Press, 2011
- Pastur, L. ; Shcherbina, M. Eigenvalue Distribution of Large Random Matrices. ( Mathematical Surveys and Monographs. Amer. Math. Soc. v. 171, 2011) 634 pp [to be published in June 2011, see http://www.ams.org/bookstore-getitem/item=SURV-171]
- Tao, T; Topics in random matrix theory.[A draft version is online at http://terrytao.files.wordpress.com/2011/02/matrix-book.pdf]