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

Abstract

The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight $\exp (-n|x{|}^{\beta })$, $\beta >0$, 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 $\beta \ge 1$.

arXiv comment56 pages

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Cited by in corpus (2)

• Emergent random matrix universality in quantum operator dynamics
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