Convergence of eigenvalues to the support of the limiting measure in critical $β$ matrix models

Abstract

We consider the convergence of the eigenvalues to the support of the equilibrium measure in the $β$ ensemble models under a critical condition. We show a phase transition phenomenon, namely that, with probability one, all eigenvalues will fall in the support of the limiting spectral measure when $β>1$, whereas this fails when $β<1$.

This is a revised version based on the comments of an anonymous referee