Crossover distributions at the edge of the rarefaction fan

TL;DR

This paper derives new crossover distributions for the KPZ equation at the edge of a rarefaction fan, connecting known distributions inside the fan to Airy processes, and establishes moment and large deviation bounds using FKG inequality.

Contribution

It introduces a new family of crossover distributions for KPZ fluctuations at the rarefaction fan edge, extending previous results and linking to continuum directed polymers.

Findings

Derived exact one-point distributions at the fan edge

Proved moment and large deviation estimates for KPZ solutions

Established FKG inequality for the stochastic heat equation

Abstract

We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_-<\rho_+$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834 (2010) 523-542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman-Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.