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 $\beta\ge 1$.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.07253
 arXiv:
 arXiv:2209.07253
 Bibcode:
 2022arXiv220907253C
 Keywords:

 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Complex Variables;
 Mathematics  Probability;
 30E05;
 30E15;
 30E20;
 30E25;
 30E99
 EPrint:
 56 pages