Stationary measures of the KPZ equation on an interval from Enaud-Derrida's matrix product ansatz representation

TL;DR

This paper derives the stationary measures of the KPZ equation on an interval by taking the weak asymmetry limit of the matrix product ansatz for the ASEP, connecting discrete models to continuum quantum mechanics.

Contribution

It provides a direct derivation of the KPZ stationary measures using the matrix product ansatz, linking discrete exclusion processes to continuum path integrals.

Findings

Derivation of KPZ stationary measures from ASEP matrix product ansatz.

Connection between discrete exclusion processes and Liouville quantum mechanics.

Recovery of recent formulas for KPZ stationary measures.

Abstract

The stationary measures of the Kardar-Parisi-Zhang equation on an interval have been computed recently. We present a rather direct derivation of this result by taking the weak asymmetry limit of the matrix product ansatz for the asymmetric simple exclusion process. We rely on the matrix product ansatz representation of Enaud and Derrida, which allows to express the steady-state in terms of re-weighted simple random walks. In the continuum limit, its measure becomes a path integral (or re-weighted Brownian motion) of the form encountered in Liouville quantum mechanics, recovering the recent formula.