The random matrix hard edge: rare events and a transition

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