Convergence of exclusion processes and KPZ equation to the KPZ fixed point

TL;DR

This paper proves that under specific scaling, the height functions of exclusion processes and the KPZ equation converge to the KPZ fixed point, linking discrete models to universal continuous limits.

Contribution

It establishes convergence of finite range exclusion processes and the KPZ equation to the KPZ fixed point under 1:2:3 scaling, extending previous results and providing new initial data classes.

Findings

Convergence of exclusion processes to the KPZ fixed point.

Convergence of the KPZ line ensemble to the Airy line ensemble.

Extension to a broader class of initial data for the KPZ equation.

Abstract

We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the KPZ equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of \cite{wu},\cite{DM20}, the KPZ line ensemble converges to the Airy line ensemble. For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of height functions with $h(x)\le C(1+|x|)$. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts. The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates.