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16
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 29 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
Large deviations for functions of two random projection matrices
 Acta Sci. Math. (Szeged
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Traces in Twodimensional QCD: The LargeN Limit, to appear
 in Traces in Geometry, Number Theory, and Quantum
"... Abstract. An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the largeN limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops. 1. ..."
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Cited by 11 (3 self)
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Abstract. An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the largeN limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops. 1.
LARGE DEVIATIONS FOR RANDOM MATRIX ENSEMBLES IN MESOSCOPIC PHYSICS
, 2007
"... In his seminal 1962 paper on the “threefold way”, Freeman Dyson classified the spaces of matrices that support the random matrix ensembles deemed relevant from the point of view of classical quantum mechanics. Recently, Heinzner, Huckleberry and Zirnbauer have obtained a similar classification base ..."
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Cited by 7 (3 self)
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In his seminal 1962 paper on the “threefold way”, Freeman Dyson classified the spaces of matrices that support the random matrix ensembles deemed relevant from the point of view of classical quantum mechanics. Recently, Heinzner, Huckleberry and Zirnbauer have obtained a similar classification based on less restrictive assumptions, thus taking care of the needs of modern mesoscopic physics. Their list is in onetoone correspondence with the infinite families of Riemannian symmetric spaces as classified by Cartan. The present paper develops the corresponding random matrix theories, with a special emphasis on large deviation principles.
An Introduction to Random
 Matrices, volume118ofCambridge Studies in Advanced Mathematics
, 2010
"... matrices and enumeration of maps ..."
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Tail inequalities for sums of random matrices that depend on the intrinsic dimension
, 2012
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On the number of minima of a random polynomial ∗
, 2007
"... We give an upper bound in O(d (n+1)/2) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d (n+1)/2)Pn, where Pn is the (unknown) measure of the set of symmetric positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give a ..."
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Cited by 2 (0 self)
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We give an upper bound in O(d (n+1)/2) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d (n+1)/2)Pn, where Pn is the (unknown) measure of the set of symmetric positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions. 1
Notes on microstate free entropy of projections
, 2006
"... We study the microstate free entropy χproj(p1,..., pn) of projections, and establish its basic properties similar to the selfadjoint variable case. Our main contribution is to characterize the pairblock freeness of projections by the additivity of χproj (Theorem 4.1), in the proof of which a tra ..."
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We study the microstate free entropy χproj(p1,..., pn) of projections, and establish its basic properties similar to the selfadjoint variable case. Our main contribution is to characterize the pairblock freeness of projections by the additivity of χproj (Theorem 4.1), in the proof of which a transportation cost inequality plays an important role. We also briefly discuss the free pressure in relation to χproj.
Concentration of the spectral measure of large Wishart matrices with dependent entries
, 2008
"... We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered. 1 ..."
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We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also considered. 1
Limit Theorems for BetaJacobi Ensembles
"... Abstract For a large betaJacobi ensemble determined by several parameters, under certain restrictions among them, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of th ..."
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Abstract For a large betaJacobi ensemble determined by several parameters, under certain restrictions among them, we obtain both the bulk and the edge scaling limits. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the Central Limit Theorems of the eigenvalues, and the limiting distributions of the empirical distributions of the eigenvalues. 1