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

Abstract

We consider the Gaussian beta-ensemble when $\beta$ scales with $n$ the number of particles such that $n^{-1}\ll \beta \ll 1$. 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.

16 pages