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

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 continuous functions with $|h(x)|\le C(1+\sqrt{|x|})$ for some $C>0$, or one narrow wedge. 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. Just before posting the first version of this article, we learned that, \emph{independently and at the same time and place}, Balint Virag found a completely different proof of the convergence of the KPZ equation to the KPZ fixed point. The methods invite extensions in different directions and it will be very interesting to see how this plays out.