Random matrix theory

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Curator: Yan Fyodorov

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.



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.


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Recommended Reading

  1. 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
  2. Di Francesco, P. ; Ginsparg, P ; Zinn-Justin, J. 2D gravity and random matrices. Phys. Rep. v. 254 (1995) no. 1-2, 1-133
  3. Beenakker C.W.J. Random-matrix theory of quantum transport. Rev. Mod. Phys. v.69 (1997) No. 3, 731-808
  4. Guhr, T.; Mueller-Groeling A.; Weidenmueller H.A. Random Matrix Theories in Quantum Physics: Common Concepts. Phys.Rep. v.299 (1998) pp 189-425
  5. Katz, N.M.; Sarnak, P. Random Matrices, Frobenius Eigenvalues, and Monodromy (Colloquium Publications . Amer Mathematical Soc. 1998), 419 pp
  6. Deift, P. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach (Courant Lecture Notes; Amer Mathematical Soc. 2000) 261 pp.
  7. Mehta, M.L. Random matrices, Third edition. Pure and Applied Mathematics (Amsterdam), 142. Elsevier/Academic Press, Amsterdam, 2004. xviii+688 pp
  8. Tulino, A. ; Verdu, S. Random Matrix Theory and Wireless Communications. Foundations and Trends in Communications and Information Theory. Now Publishers Inc. (2004) 192 pp
  9. 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
  10. Hiai F. and Petz D. The Semicircle Law, Free Random Variables and Entropy (Mathematical Surveys and Monographs. Amer Math. Soc. 2007) 376pp
  11. Anderson, G.W ; Guionnet, A; Zeitouni O. An Introduction to Random Matrices. Studies in Advanced Mathematics, v. 118, Cambridge University Press, 2009
  12. Deift. P; Gioev D. Random Matrix Theory: Invariant Ensembles and Universality. (Courant Lecture Notes; Amer Mathematical Soc. 2009) 217 pp.
  13. Bai, Z.D.; Silverstein J. W. Spectral analysis of Large Dimensional Random Matrices. (Springer Series in Statistics, 2nd ed., 2010), 552 pp
  14. Forrester, P.J. Log-Gases and Random Matrices (LMS-34) (London Mathematical Society Monographs), Princeton University Press, 2010, 808 pp
  15. 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
  16. Oxford Handbook on Random Matrix theory, edited by Akemann G.; Baik, J. ; Di Francesco P. , Oxford University Press, 2011
  17. 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]
  18. Tao, T; Topics in random matrix theory.[A draft version is online at http://terrytao.files.wordpress.com/2011/02/matrix-book.pdf]
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