Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case
Michel Ledoux, Université Toulouse
Abstract
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.
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