Large deviations of crowding in finite $\beta$-ensembles

TL;DR

This paper investigates large deviation principles for the number of points in a subset of finite $eta$-ensembles on the real line or complex plane, establishing bounds with speed $n^2$ and identifying the rate function.

Contribution

It provides a direct method to prove large deviation bounds for finite $eta$-ensembles in the complex plane, where the contraction principle is insufficient.

Findings

Large deviation bounds with speed $n^2$ for point counts in $eta$-ensembles.

Explicit characterization of the rate function for the large deviations.

Methodology applicable to complex plane ensembles where standard contraction does not apply.

Abstract

We consider finite $\beta$-ensembles $\mathcal X_{n,\beta}^{\mathbb F}$ with $n$ points on $\mathbb F$, where $\mathbb F$ denotes either the real line or the complex plane. Let $U$ be a bounded subset of $ \mathbb F$ such that $\partial U$ (the boundary of $U$) is polar for $\mathbb F=\mathbb R$ and $\partial U$ is a closed $1$--rectifiable set with finite $1$-dimensional Hausdorff measure. Suppose $\mathcal X_{n,\beta}^{\mathbb F}(U)$ denotes the number of points in the region $U$. We show that the sequence of laws of $\{n^{-1}\mathcal X_{n,\beta}^{\mathbb F}(U); n\ge 1\}$ satisfies the large deviation type bound with speed $n^2$ and with a good rate function. For $\mathbb{F} = \mathbb{R}$, this result can be derived using the contraction principle. However, when $\mathbb{F} = \mathbb{C}$, the contraction principle does not yield the desired outcome. Therefore, we adopt a direct approach to establish our results.