  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.
