LA Probability Forum

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 particles density. The main goal of this article is to analyse the interacting particle system using the framework of regularity structures without applying the Gärtner transformation, a discrete version of the Cole-Hopf transformation which linearises the KPZ equation. 

Our analysis relies on discretisation framework for regularity structures developed by Erhard and Hairer [AIHP 2019] as well as estimates for iterated integrals with respect to jump martingales derived by Grazieschi, Matetski and Weber [PTRF 2025]. The main technical challenge addressed in this work is the renormalisation procedure which requires a subtle analysis of regularity preserving discrete convolution operators.