A large deviation principle for the Gaussian beta ensemble in the high temperature regime

Abstract

We show a large deviation principle for the weighted spectral measure of the Gaussian beta ensemble in the high temperature regime, when $n{\beta }_n\to 2c\in [0,\infty )$. The resulting rate function has a novel and interesting structure: it is finite only for discrete measures and moreover, in the very high-temperature regime (when $c=0$), additional terms may arise that penalize each support point beyond the first.