A Riemann-Hilbert approach to the lower tail of the KPZ equation

TL;DR

This paper develops a Riemann-Hilbert approach to analyze the lower tail behavior of the KPZ equation, providing precise asymptotics and connecting Fredholm determinants to integrable systems.

Contribution

It introduces a novel Riemann-Hilbert framework to derive asymptotics of Fredholm determinants related to the KPZ equation's lower tail.

Findings

Derived asymptotics for Fredholm determinants

Established a Riemann-Hilbert representation of derivatives

Refined lower tail asymptotics for KPZ with narrow wedge

Abstract

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. We show that logarithmic derivatives of the Fredholm determinants can be expressed in terms of a 2x2 Riemann-Hilbert problem, and we use this to derive asymptotics for the Fredholm determinants. As an application of our result, we derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal.