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From Gumbel to TracyWidom
"... Abstract. The TracyWidom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp( − exp(−x)), the Gumbel distribution and the TracyWidom distribution. There is a family ..."
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Abstract. The TracyWidom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution exp( − exp(−x)), the Gumbel distribution and the TracyWidom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density exp(−x) and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, M = M0 + √ 2SV, with M0 diagobal with independent elements and V from GUE. Here we do not see a transition from TracyWidom to Gumbel, but rather a transition from TracyWidom to Gaussian. 1. Introduction and
The moments of N ×N Hermitian random matrices HN are given by expression
, 2008
"... We describe an elementary method to get nonasymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. We derive a system of recurrent relations for the moments and the covariance terms and develop a triangular scheme to prove the recu ..."
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We describe an elementary method to get nonasymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. We derive a system of recurrent relations for the moments and the covariance terms and develop a triangular scheme to prove the recurrent estimates. The estimates we obtain are asymptotically exact in the sense that they give exact expressions for the first terms of 1/Nexpansions of the moments and covariance terms. As the basic example, we consider the Gaussian Unitary Ensemble of random matrices (GUE). Immediate applications include Gaussian Orthogonal Ensemble and the ensemble of Gaussian antisymmetric Hermitian matrices. Finally we apply our method to the ensemble of N ×N Gaussian Hermitian random matrices H (N,b) whose elements are zero outside of the band of width b. The other elements are taken from GUE; the matrix obtained is normalized by b −1/2. We derive the estimates for the moments
Real symmetric random matrices and paths counting
, 2004
"... Exact evaluation of Tr < S p> is here performed for real symmetric matrices S of arbitrary order n, where the entries are independent identically distributed random variables, with an arbitrary probability distribution, up to some integer p. These polynomials provide useful information on the ..."
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Exact evaluation of Tr < S p> is here performed for real symmetric matrices S of arbitrary order n, where the entries are independent identically distributed random variables, with an arbitrary probability distribution, up to some integer p. These polynomials provide useful information on the spectral density of the ensemble in the large n limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large n limit. 1
Limit Distributions for Random Hankel, Toeplitz Matrices and Independent Products
, 2009
"... For random selfadjoint (real symmetric, complex Hermitian, or quaternion selfdual) Toeplitz matrices and real symmetric Hankel matrices, the existence of universal limit distributions for eigenvalues and products of several independent matrices is proved. The joint moments are the integral sums rel ..."
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For random selfadjoint (real symmetric, complex Hermitian, or quaternion selfdual) Toeplitz matrices and real symmetric Hankel matrices, the existence of universal limit distributions for eigenvalues and products of several independent matrices is proved. The joint moments are the integral sums related to certain pair partitions. Our method can apply to random Hankel and Toeplitz band matrices, and the similar results are given. In particular, when the band width grows slowly as the dimension N → ∞, the exact limit distribution functions are given (N(0, 1) for Toeplitz band matrices) and some asymptotic commutativity is observed. 1