# Fredholm determinant solutions of the Painlev\'e II hierarchy and gap probabilities of determinantal point processes

## TL;DR

This paper establishes a connection between Fredholm determinants with special kernels and solutions to the Painlevé II hierarchy, providing asymptotic analysis and confirming recent conjectures in random matrix theory and statistical physics.

## Contribution

It generalizes a recent conjecture by linking Fredholm determinants to Painlevé II hierarchy solutions and derives their asymptotic behaviors.

## Key findings

- Fredholm determinants relate to Painlevé II hierarchy solutions
- Asymptotics at infinity for Painlevé transcendents derived
- Large gap asymptotics for determinantal point processes obtained

## Abstract

We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlev\'e II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm\infty$ for the Painlev\'e transcendents and large gap asymptotics for the corresponding point processes.