Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime

TL;DR

This paper analyzes the asymptotic behavior of linear eigenvalue statistics in $eta$ matrix models within the multi-cut regime, providing detailed expansions and limit results including expectation and variance, with quasi-periodic terms.

Contribution

It offers the first detailed asymptotic expansion up to $O(n^{-1})$ for the partition function in the multi-cut regime of $eta$ matrix models, including explicit formulas for eigenvalue statistics.

Findings

Derived the limit of the generating functional of linear eigenvalue statistics.

Obtained explicit expressions for expectation and variance of eigenvalue statistics.

Identified quasi-periodic in $n$$ terms in the general case.

Abstract

We study the asymptotic expansion in $n$ for the partition function of $\beta$ matrix models with real analytic potentials in the multi-cut regime up to the $O(n^{-1})$ terms. As a result, we find the limit of the generating functional of linear eigenvalue statistics and the expressions for the expectation and the variance of linear eigenvalue statistics, which in the general case contain the quasi periodic in $n$ terms.