Replica approach to the KPZ equation with half Brownian motion initial condition

TL;DR

This paper analyzes the KPZ equation with half Brownian motion initial condition using the replica Bethe ansatz, deriving a Fredholm determinant expression for the height distribution and exploring its asymptotics and multi-point correlations.

Contribution

It introduces a replica Bethe ansatz approach to derive a Fredholm determinant formula for the KPZ height distribution with half Brownian initial condition, advancing analytical understanding.

Findings

Height distribution expressed as a Fredholm determinant.

Asymptotic behavior of the height distribution analyzed.

Multi-point height distribution discussed.

Abstract

We consider the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with half Brownian motion initial condition, studied previously through the weakly asymmetric simple exclusion process. We employ the replica Bethe ansatz and show that the generating function of the exponential moments of the height is expressed as a Fredholm determinant. From this the height distribution and its asymptotics are studied. Furthermore using the replica method we also discuss the multi-point height distribution. We find that some nice properties of the deformed Airy functions play an important role in the analysis.