Crossover to the KPZ equation

We characterize the crossover regime to the KPZ equation for a class of one-dimensional weakly asymmetric exclusion processes. The crossover depends on the strength asymmetry $an^{2-\gamma}$ ($a,\gamma>0$) and it occurs at $\gamma=1/2$. We show that the density field is a solution of an Ornstein-Uhlenbeck equation if $\gamma\in(1/2,1]$, while for $\gamma=1/2$ it is an energy solution of the KPZ equation. The corresponding crossover for the current of particles is readily obtained.