THE CROSSOVER TO THE KPZ EQUATION

ABSTRACT. In this paper we consider the one-dimensional weakly asymmet-ric simple exclusion process under the invariant state νρ: the Bernoulli prod-uct measure of parameter ρ ∈ (0, 1). We show that the limit density field is governed by an Ornstein-Uhlenbeck process for strength asymmetry n2−γ if γ ∈ (1/2, 1), while for γ = 1/2 it is an energy solution of the KPZ equation. From this result we obtain that the fluctuations of the current of particles are Gaussian distributed for γ ∈ (1/2, 1), while for γ = 1/2 the limit distribution is written in terms of the KPZ equation. 1