Coulomb-gas electrostatics controls large fluctuations of the KPZ equation

TL;DR

This paper establishes a large deviation principle for the KPZ equation's height distribution, linking it to Coulomb-gas electrostatics via the Airy point process, and provides finite-time tail bounds demonstrating a crossover in decay behavior.

Contribution

The authors develop a Coulomb-gas electrostatic framework to derive the KPZ large deviation rate function directly from the Airy point process, extending to half-space cases and finite-time tail bounds.

Findings

Large deviation principle for KPZ height distribution established.

Explicit rate function derived using electrostatic analogy.

Finite-time tail bounds show crossover in decay exponents.

Abstract

We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process -- an infinite particle Coulomb-gas which arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straight-forward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution which demonstrate a crossover between exponential decay with exponent $3$ (in the shallow left tail) to exponent $5/2$ (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech, 063203 (2017)] via a WKB approximation analysis of a non-local, non-linear integro-differential equation generalizing Painlev\'e II which Amir et al. [Comm. Pure Appl. Math. 64, 466 (2011)] related to the KPZ one-point distribution.