Weakly Asymmetric Non-Simple Exclusion Process and the Kardar-Parisi-Zhang Equation

TL;DR

This paper proves the convergence of a class of non-simple exclusion processes with limited hopping range to the KPZ equation, establishing a new universality result in interacting particle systems.

Contribution

It introduces a generalized transformation to analyze non-simple exclusion processes and proves their convergence to the KPZ equation for hopping range up to 3, a novel universality result.

Findings

Proved convergence of exclusion processes to KPZ equation for hopping range ≤ 3.

Derived the exact one-point distribution for the step initial condition.

Established the first universality result for non-solvable interacting particle systems.

Abstract

We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing Gaertners Cole-Hopf transformation. We identify the main non-linearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar-Parisi-Zhang (KPZ) equation. This is the first universality result concerning interacting particle systems in the context of KPZ universality class. While this class of exclusion processes are not explicitly solvable, we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (2011) and our convergence result.