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 vertical bar x vertical bar(beta)), beta > 0, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in beta. If beta >= 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 >= 1.