Ergodicity of the KPZ Fixed Point

TL;DR

This paper proves that the KPZ fixed point, modeling height fluctuations in TASEP, converges to a two-sided Brownian motion, establishing ergodicity and invariant measures using coupling methods.

Contribution

It demonstrates the ergodicity of the KPZ fixed point and identifies its invariant measure as a two-sided Brownian motion with zero drift.

Findings

KPZ fixed point converges to its invariant measure

Invariant measure is a two-sided Brownian motion with zero drift

Coupling method effectively compares TASEP height function with invariant process

Abstract

The Kardar-Parisi-Zhang (KPZ) fixed point is a Markov process, recently introduced by Matetski, Quastel, Remenik (arXiv:1701.00018), that describes the limit fluctuations of the height function associated to the totally asymmetric simple exclusion process (TASEP), and it is conjectured to be at the centre of the KPZ universality class. Our main result is that the KPZ incremental process converges weakly to its invariant measure, given by a two-sided Brownian motion with zero drift and diffusion coefficient 2. The heart of the proof is the coupling method that allows us to compare the TASEP height function with its invariant process, which under the KPZ scaling turns into uniform estimates for the KPZ fixed point.