Weak and strong confinement in the Freud random matrix ensemble and gap probabilities

The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight $\exp (-n\mathrm{\mid }x{\mathrm{\mid }}^{\beta })$, $\beta >0$, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in $\beta$. If $\beta \ge 1$, it is described by the standard sine process. Below the critical value $\beta =1$, it is described by a process depending on the value of $\beta$, and we determine the first two terms of the large gap probability in it. This so called weak confinement range $0<\beta <1$ corresponds to the Freud weight with the indeterminate moment problem. We also find the multiplicative constant in the asymptotic expansion of the Freud multiple integral for $\beta \ge 1$.