Around the same time the statistics of the largest eigenvalue in GaussianEnsembles 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-tailedrandom 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.
An Introduction To Random Matrices (Cambridge Studies In Advanced Mathematics) Book Pdf
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