On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice

TL;DR

This paper analyzes the mean density of states for certain beta-related random matrix models in the high temperature regime, providing explicit formulas and applications to the Toda lattice's Lax matrix.

Contribution

It introduces and studies Gaussian, Laguerre, and Jacobi alpha-ensembles in the high temperature limit, deriving explicit mean densities of states and applying results to the Toda lattice.

Findings

Proved convergence of empirical spectral distributions to explicit mean densities.

Computed the mean density of states for the Toda lattice Lax matrix.

Established explicit formulas for densities in beta-ensembles at high temperature.

Abstract

In this manuscript we study tridiagonal random matrix models related to the classical $\beta$-ensembles (Gaussian, Laguerre, Jacobi) in the high temperature regime, i.e. when the size $N$ of the matrix tends to infinity with the constraint that $\beta N=2\alpha$ constant, $\alpha > 0$. We call these ensembles the Gaussian, Laguerre and Jacobi $\alpha$-ensembles and we prove the convergence of their empirical spectral distributions to their mean densities of states and we compute them explicitly. As an application we explicitly compute the mean density of states of the Lax matrix of the Toda lattice with periodic boundary conditions with respect to the Gibbs ensemble.