Edgeworth Type Expansion of the Distribution of the Largest Eigenvalue in Classical Random Matrix Ensembles

Dr. Leonard N. Choup

Department of Mathematical Sciences
University of Alabama in Huntsville

August 24, 2007
202 Madison Hall
3:00 PM (Refreshments at 2:30 in 201 Madison Hall)

Abstract

One part of Random Matrix Theory (RMT) deals with the statistical properties of eigenvalues of large N × N matrices from random matrix ensembles. Of interest is the distribution of eigenvalues. In the 50's it was conjectured and later proved that, under a proper scaling the bulk of the eigenvalues is predicted to sit on the upper half semicircle (Wigner semicircle law).

The behavior of the eigenvalues outside the bulk of the spectrum was an open problem until 1994. The limiting distribution of the largest eigenvalue when the size N of the matrices goes to infinity was derived by Craig A. Tracy and Harold Widom (Tracy-Widom distributions).

In practice one would like to know this distribution for finite N not the limiting distribution. This presentation outlines the steps needed to derive the distribution of the largest eigenvalue for finite N.

We derive an edge scaling correction for Gaussian and Laguerre kernels for Unitary Ensembles, derive an expansion of the probability distribution function of the largest eigenvalue F_N;2(t)= P₂(λ_{2, max} = t) in terms ofN and the Tracy-Widom distribution F₂, then give an outline of the steps needed to extend this analysis to the orthogonal and symplectic matrix ensembles.

Keywords: Integral operator, Trace Class operator, Tracy-Widom distribution.