Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ

TL;DR

This paper derives uniform asymptotic formulas for Airy kernel determinants linked to free fermion models and KPZ equation solutions, providing insights into initial data and tail behaviors of these complex systems.

Contribution

It introduces uniform asymptotics for deformed Airy kernel determinants, connecting them to KdV solutions and KPZ tail asymptotics, advancing understanding of these integrable models.

Findings

Uniform asymptotics for Airy kernel determinants obtained

Provides uniform initial data for KdV solutions

Establishes uniform lower tail asymptotics for KPZ narrow wedge

Abstract

We obtain uniform asymptotics for deformed Airy kernel determinants, which arise in models of finite temperature free fermions and which characterize the narrow wedge solution of the Kardar-Parisi-Zhang equation. The asymptotics for the determinants yield uniform initial data for an associated family of solutions to the Korteweg-de Vries equation, and uniform lower tail asymptotics for the narrow wedge solution of the Kardar-Parisi-Zhang equation.