Large Deviations for Random Spectral Measures and Sum Rules

TL;DR

This paper establishes a Large Deviation Principle for random spectral measures in GUE and related ensembles, linking the rate function to information measures and extreme eigenvalue behavior, with applications to Laguerre and Jacobi ensembles.

Contribution

It introduces a novel Large Deviation Principle for spectral measures in GUE and extends the approach to other classical ensembles, connecting rate functions to spectral properties.

Findings

Rate function includes reversed Kullback-Leibler divergence for the absolutely continuous part.

Singular part rate function relates to extreme eigenvalue large deviations.

Method applies to Laguerre and Jacobi ensembles, with less explicit rate functions.

Abstract

We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\beta$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit.