Large Deviations Principle for the Largest Eigenvalue of the Gaussian $$\beta $$β-Ensemble at High Temperature

Abstract

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