The general beta hard edge

Wednesday, June 20, 2012 - 2:00pm - 3:15pm
Lind 305
Brian Rider (University of Colorado)
In RMT the hard edge refers to the scaling limits of the minimal eigenvalues for matrices of sample covariance type. In the classical invariant ensembles, the limit distributions are characterized by a Bessel kernel and an associated Painleve III equation (as opposed to the better known Airy kernel and Painleve II descriptions at the soft edge). We will show that in the general beta setting these descriptions can be replaced by a limiting (random) differential operator and/or the hitting distributions of a related diffusion process. With this in hand various applications of the operator/diffusion picture will be presented, such as sharp tail asymptotics for the beta hard edge laws, the effects of finite rank perturbations to the input matrices, and the link to Painleve III.

Joint work with J. Ramirez and I. Rumanov.
MSC Code: