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

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