The random matrix hard edge: rare events and a transition

TL;DR

This paper investigates rare events and phase transitions at the hard edge of random matrix spectra, revealing asymptotic behaviors and connections to the Sine$_eta$ process.

Contribution

It introduces new asymptotic results for the hard edge process, including a transition to bulk behavior and comparisons to the Sine$_eta$ process.

Findings

Established a central limit theorem for the number of points in large intervals.

Derived large deviation principles for rare events.

Demonstrated a transition from hard edge to bulk behavior.

Abstract

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 $\lambda \to \infty$. 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.