Asymptotics of the Smallest Eigenvalue Distributions of Freud Unitary Ensembles

2024/02/23 by Chao Min, Liwei Wang, Min, Chao +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Physical sciences #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.15190

openalex publication_date 2024/02/23 · openalex created_date 2024/02/27 · openalex updated_date 2026/08/01

Abstract

We consider the smallest eigenvalue distributions of some Freud unitary ensembles, that is, the probabilities that all the eigenvalues of the Hermitian matrices from the ensembles lie in the interval (t,∞). This problem is related to the Hankel determinants generated by the Freud weights with a jump discontinuity. By using Chen and Ismail's ladder operator approach, we obtain the discrete systems for the recurrence coefficients of the corresponding orthogonal polynomials. This enables us to derive the large n asymptotics of the recurrence coefficients via Dyson's Coulomb fluid approach. We finally obtain the large n asymptotics of the Hankel determinants and that of the probabilities from their relations to the recurrence coefficients and with the aid of some recent results in the literature.

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