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

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.

v2: Added Section 2.4 about summation over the zero mode. 14 pages

References in corpus (4)

• Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
• Markov processes related to the stationary measure for the open KPZ equation
• Continued Fractions and the Partially Asymmetric Exclusion Process
• Stationary measure for the open KPZ equation

Cited by in corpus (6)

• Approximating the stationary distribution of the ASEP with open boundaries
• KPZ fluctuations in finite volume
• A two-line representation of stationary measure for open TASEP
• Limit theorems for random Motzkin paths near boundary
• Limit fluctuations of stationary measure of totally asymmetric simple exclusion process with open boundaries on the coexistence line
• Limits of Random Motzkin paths with KPZ related asymptotics