Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlev\'e II system

TL;DR

This paper derives Tracy-Widom formulas for Fredholm determinants associated with Painlevé kernels in critical unitary ensembles, linking them to coupled Painlevé II systems and providing explicit large gap asymptotics involving the Riemann zeta-function.

Contribution

It provides explicit Tracy-Widom formulas for Fredholm determinants in critical ensembles using coupled Painlevé II solutions, and computes their large gap asymptotics with explicit constants.

Findings

Explicit Tracy-Widom formulas for Fredholm determinants.

Large gap asymptotics with constants involving Riemann zeta-function.

Connection between eigenvalue gap probabilities and coupled Painlevé II systems.

Abstract

We study Fredholm determinants of the Painlev\'e II and Painlev\'e XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlev\'e II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.