Fredholm Determinants from Schr\"odinger Type Equations, and Deformation of Tracy-Widom Distribution

TL;DR

This paper generalizes the Tracy-Widom distribution by analyzing Fredholm determinants derived from kernels satisfying Schr"odinger type equations, connecting them to isomonodromic systems and broadening their applications.

Contribution

It introduces a new class of Fredholm determinants based on Schr"odinger equations, extending the Tracy-Widom distribution to more general defining functions.

Findings

Generalized Tracy-Widom distribution for a class of Schr"odinger-based kernels

Established connection between Fredholm determinants and isomonodromic systems

Provided a framework for broader applications of these distributions

Abstract

We undertake an analysis of Fredholm determinants arising from kernels whose defining functions satisfy a Schr\"odinger type equation. When this defining function is the Airy one, the evaluation of the corresponding Fredholm determinant yields the notorious Tracy-Widom distribution [hep-th/9211141], which has found many applications in numerous domains. In this paper, we unveil a generalization of the Tracy-Widom distribution for a generic class of defining functions. Furthermore, we bring forth a direct application of our upshot and survey the relation between the framework which we employ and isomonodromic systems.