Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case

By Michel Ledoux. 2004

69 citations recorded by the source corpus.

Following the investigation by U. Haagerup and S. Thorbjornsen, we present a simple differential approach to the limit theorems for empirical spectral distributions of complex random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles. In the framework of abstract Markov diffusion operators, we derive by the integration by parts formula differential equations for Laplace transforms and recurrence equations for moments of eigenfunction measures. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. The moment recurrence relations are used to describe sharp, non asymptotic, small deviation inequalities on the largest eigenvalues at the rate given by the Tracy-Widom asymptotics.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8b789dea4ecc80aed318f695f9b9aaef16b1dfcc

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles (2009, background): “The exact limiting distribution for complex case was given explicitly in [14], see also [53].”
Deviation Inequalities on Largest Eigenvalues (2007, background): “For example, the Laguerre and Jacobi ensembles are studied along these lines in [ Le3 ] to yield deviation inequalities at the Tracy-Widom rate of the largest eigenvalue of Wishart and Beta matrices.”
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles (2006, background): “The weak convergence of the ESD ([11]) and the control on the extremal eigenvalues (Ledoux [44], Collins [13] and references therein), yield, if u′ ≥ 1”
Small deviations for beta ensembles (2009, background): “As discussed in [14, 15], the right-tail inequality for the GUE (as well as for the Laguerre Unitary Ensemble, again see below) may be shown to follow from results of Johansson [11] for a more general invariant model related to the geometric distribution that uses large deviation asymptotics and sub”