Scaling limit of a weakly asymmetric simple exclusion process in the framework of regularity structures

TL;DR

This paper proves that a rescaled weakly asymmetric exclusion process converges to the KPZ equation's solution using regularity structures, avoiding traditional Cole-Hopf transforms, and advances the mathematical understanding of particle systems.

Contribution

It introduces a novel analysis of the exclusion process via regularity structures without relying on the Gaertner transform, extending the theoretical framework.

Findings

Convergence of the rescaled exclusion process to KPZ solution established.

Development of a discretisation framework for regularity structures.

New renormalisation techniques for discrete convolution operators.

Abstract

We prove that a parabolically rescaled and suitably renormalised height function of a weakly asymmetric simple exclusion process on a circle converges to the Cole-Hopf solution of the KPZ equation. This is an analogue of the celebrated result by Bertini and Giacomin from 1997 for the exclusion process on a circle with any particle density. The main goal of this article is to analyse the interacting particle system using the framework of regularity structures without applying the Gaertner transform, a discrete version of the Cole-Hopf transform which linearises the KPZ equation. Our analysis relies on discretisation framework for regularity structures developed by Erhard and Hairer as well as estimates for iterated integrals with respect to cadlag martingales derived by Grazieschi, Matetski and Weber. The main technical challenge addressed in this work is the renormalisation procedure which requires a subtle analysis of regularity preserving discrete convolution operators.