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

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