Large deviations principle for the largest eigenvalue of the Gaussian beta-ensemble at high temperature

TL;DR

This paper establishes a large deviations principle for the largest eigenvalue of the Gaussian beta-ensemble when the inverse temperature parameter scales between 1/n and 1, revealing its probabilistic behavior at high temperature.

Contribution

It provides the first large deviations result for the largest eigenvalue in the high-temperature regime of the Gaussian beta-ensemble with explicit rate function.

Findings

Largest eigenvalue satisfies a large deviations principle with speed nβ.

Largest eigenvalue converges in probability to 2.

Explicit rate function for the large deviations.

Abstract

We consider the Gaussian beta-ensemble when $\beta$ scales with $n$ the number of particles such that $\displaystyle{{n}^{-1}\ll \beta\ll 1}$. Under a certain regime for $\beta$, we show that the largest particle satisfies a large deviations principle in $\mathbb{R}$ with speed $n\beta$ and explicit rate function. As a consequence, the largest particle converges in probability to $2$, the rightmost point of the semicircle law.