Local KPZ behavior under arbitrary scaling limits

TL;DR

This paper introduces the concept of local KPZ behavior to analyze surface growth models, showing that such behavior occurs under arbitrary scaling limits across various dimensions, including models like directed polymers.

Contribution

It formalizes local KPZ behavior and proves its occurrence under any scaling limit for a broad class of surface growth models, extending understanding beyond one dimension.

Findings

Local KPZ behavior occurs under arbitrary scaling limits.

The framework applies to models including directed polymers.

Surface growth decomposes into key terms with negligible remainders.

Abstract

One of the main difficulties in proving convergence of discrete models of surface growth to the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than one is that the correct way to take a scaling limit, so that the limit is nontrivial, is not known in a rigorous sense. To understand KPZ growth without being hindered by this issue, this article introduces a notion of "local KPZ behavior", which roughly means that the instantaneous growth of the surface at a point decomposes into the sum of a Laplacian term, a gradient squared term, a noise term that behaves like white noise, and a remainder term that is negligible compared to the other three terms and their sum. The main result is that for a general class of surfaces, which contains the model of directed polymers in a random environment as a special case, local KPZ behavior occurs under arbitrary scaling limits, in any dimension.