The random matrix hard edge: rare events and a transition

Abstract

We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval $[0,\lambda ]$ as $\lambda \to \infty $. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the $\mathrm{Sine} _\beta $ process.