Large deviations principle for some beta-ensembles

TL;DR

This paper establishes a large deviations principle for beta-ensembles derived from holomorphic sections of line bundles over complex manifolds, providing insights into their convergence behavior and connections to random matrix models.

Contribution

It introduces a large deviations principle with an effective speed for empirical measures of beta-ensembles on complex manifolds, extending understanding of their asymptotic properties.

Findings

Empirical measures converge almost surely to an equilibrium measure.

Established a large deviations principle with effective convergence speed.

Applicable to beta-ensembles on real spheres and Euclidean spaces.

Abstract

Let L be a positive line bundle over a projective complex manifold X. Consider the space of holomorphic sections of the tensor power of order p of L. The determinant of a basis of this space, together with some given probability measure on a weighted compact set in X, induces naturally a beta-ensemble, i.e., a random point process on the compact set. Physically, this general setting corresponds to a gas of free fermions in X and may admit some random matrix models. The empirical measures, associated with such beta-ensembles, converge almost surely to an equilibrium measure when p goes to infinity. We establish a large deviations principle (LDP) with an effective speed of convergence for these empirical measures. Our study covers the case of some beta-ensembles on a compact subset of a real sphere or of a real Euclidean space.