Stationary correlations for the 1D KPZ equation

TL;DR

This paper derives exact stationary properties of the 1D KPZ equation using the replica method, providing explicit formulas for height distribution and correlations in the stationary state.

Contribution

It introduces novel techniques to handle two-sided Brownian initial conditions in replica analysis, yielding explicit formulas for height distribution and correlations.

Findings

Explicit Fredholm determinant representation of height distribution

Exact space-time two-point correlation function derived

Stationary state characterized by two-sided Brownian motion initial condition

Abstract

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit representation of the probability distribution of the height in terms of the Fredholm determinants. Furthermore from this expression, we also get the exact expression of the space-time two-point correlation function.