Approach to stationarity for the KPZ fixed point with boundaries

TL;DR

This paper investigates the approach to stationarity in the KPZ fixed point with boundaries, using exact methods to analyze fluctuations in TASEP and deriving explicit expressions for late-time corrections.

Contribution

It provides the first exact expressions for late-time corrections to stationarity in KPZ with boundaries, including special cases with explicit formulas.

Findings

Derived exact late-time correction formulas for KPZ with boundaries.

Connected TASEP fluctuations to extreme value statistics of Brownian paths.

Obtained explicit solutions for stationary and narrow wedge initial conditions.

Abstract

Current fluctuations for the one-dimensional totally asymmetric exclusion process (TASEP) connected to reservoirs of particles, and their large scale limit to the KPZ fixed point in finite volume, are studied using exact methods. Focusing on the maximal current phase for TASEP, corresponding to infinite boundary slopes for the KPZ height field, we obtain for general initial condition an exact expression for the late time correction to stationarity, involving extreme value statistics of Brownian paths. In the special cases of stationary and narrow wedge initial conditions, a combination of Bethe ansatz and numerical conjectures alternatively provide fully explicit exact expressions.