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 |β),β >0 , on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in β . If β ≥1 , it is described by the standard sine process. Below the critical value β =1 , it is described by a process depending on the value of β , and we determine the first two terms of the large gap probability in it. This so called weak confinement range 0 <β <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 β ≥1.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- August 2023
- DOI:
- 10.1007/s00220-023-04749-y
- arXiv:
- arXiv:2209.07253
- Bibcode:
- 2023CMaPh.402..833C
- Keywords:
-
- Mathematical Physics;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Complex Variables;
- Mathematics - Probability;
- 30E05;
- 30E15;
- 30E20;
- 30E25;
- 30E99
- E-Print:
- 56 pages