Weak and Strong Confinement in the Freud Random Matrix Ensemble and Gap Probabilities

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