Convergence of eigenvalues to the support of the limiting measure in critical $\beta$ matrix models

We consider the convergence of the eigenvalues to the support of the equilibrium measure in the $\beta$ ensemble models under a critical condition. We show a phase transition phenomenon, namely that, with probability one, all eigenvalues will fall in the support of the limiting spectral measure when $\beta >1$, whereas this fails when $\beta <1$.