A unified fluctuation formula for one-cut $\beta$-ensembles of random matrices

TL;DR

This paper derives a universal fluctuation formula for covariances of power traces in one-cut $eta$-ensembles of random matrices, applicable to classical models and revealing connections to planar map enumeration.

Contribution

It provides a unified, closed-form fluctuation formula for one-cut $eta$-ensembles, encompassing classical models and highlighting universality.

Findings

Derived a universal generating function for covariances in one-cut $eta$-ensembles

Unified treatment of $eta$-Gaussian, $eta$-Wishart, and $eta$-Jacobi ensembles

Connected trace fluctuations to planar map enumeration

Abstract

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $\beta$-ensemble. As particular cases of the main result we consider the classical $\beta$-Gaussian, $\beta$-Wishart and $\beta$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.