The crossover regime for the weakly asymmetric simple exclusion process

TL;DR

This paper investigates the fluctuation behavior of the current in the weakly asymmetric simple exclusion process, revealing a crossover from Gaussian to Tracy-Widom distributions under specific scaling regimes.

Contribution

It derives the limiting distribution of the integrated current in WASEP during the crossover regime, connecting Gaussian and Tracy-Widom fluctuations.

Findings

Distribution converges to Tracy-Widom for large times

Identifies the crossover scale for asymmetry and time

Provides integral representation involving Fredholm determinants

Abstract

We consider the asymmetric simple exclusion process in one dimension with weak asymmetry (WASEP) and 0-1 step initial condition. Our interest are the fluctuations of the time-integrated particle current at some prescribed spatial location. One expects a crossover from Gaussian to Tracy-Widom distributed fluctuations. The appropriate crossover scale is an asymmetry of order $\epsilon^{-1/2}$, times of order $\epsilon^{-2}$, and a spatial location of order $\epsilon^{-3/2}$. For this parameter window we obtain the limiting distribution function of the integrated current in terms of an integral over the difference of two Fredholm determinants. For large times, on the scale $\epsilon^{-2}$, this distribution function converges to the one of Tracy-Widom.