Asymptotic expansion of beta matrix models in the multi-cut regime

TL;DR

This paper develops a detailed asymptotic expansion for beta matrix models in the multi-cut regime, analyzing fluctuations and applying results to Toda chain solutions and orthogonal polynomials.

Contribution

It provides the first comprehensive asymptotic expansion in the multi-cut regime for beta matrix models, including fluctuation analysis and applications to integrable systems.

Findings

Established $1/N$ expansion for fixed filling fractions.

Characterized fluctuations as sums of Gaussian variables.

Applied results to Toda chain asymptotics and orthogonal polynomials.

Abstract

We establish the asymptotic expansion in $\beta$ matrix models with a confining, off-critical potential, in the regime where the support of the equilibrium measure is a union of segments. We first address the case where the filling fractions of these segments are fixed, and show the existence of a $1/N$ expansion. We then study the asymptotics of the sum over the filling fractions, to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ($\beta = 2$) as well as orthogonal ($\beta = 1$) and skew-orthogonal ($\beta = 4$) polynomials outside the bulk.