On the mean density of states of some matrices related to the beta ensembles and an application to the Toda lattice

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