Higher order analogues of the Tracy-Widom distribution and the Painleve II hierarchy

2009/01/16 by T. Claeys, Tom Claeys, A. Its +6 · 2 citations
Mathematics · Physics and Astronomy · #33E17 #35Q15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Random Matrices and Applications #Statistical Distribution Estimation and Applications #math-ph #math.CA #math.MP #msc:33E17 #msc:35Q15

paper · pdf · doi:10.48550/arxiv.0901.2473

45 pages

arxiv created 2009/01/16 · openalex publication_date 2009/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are built out of functions associated with the Painlevé I hierarchy. The Fredholm determinants related to those kernels are higher order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painlevé II hierarchy. In addition we compute large gap asymptotics for the Fredholm determinants.

Cited by

• The Painlevé I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy
• Unitary ensembles with a critical edge point, their multiplicative statistics and the Korteweg-de-Vries hierarchy

Related

• The multiplicative constant in asymptotics of higher-order analogues of the Tracy-Widom distribution
• Numerical study of higher order analogues of the Tracy-Widom distribution
• Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential
• The generating function for the Airy point process and a system of coupled Painlevé II equations
• Asymptotics of a Fredholm determinant involving the second Painlevé transcendent