Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential

2017/10/23 by Dan Dai, Dai, Dan, Shuai‐Xia Xu +3 · 1 citation
Mathematics · #33E17 #34M55 #41A60 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1710.08132

openalex publication_date 2017/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Fredholm determinant of an integrable operator acting on the interval (0,s) whose kernel is constructed out of a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order k in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large s asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to the coupled Painlevé III system.

Citations

• Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

Cited by

• Orthogonal polynomials for the singularly perturbed Laguerre weight, Hankel determinants and asymptotics

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