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The crossover regime for the weakly asymmetric simple exclusion process

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 $ε^{-1/2}$, times of order $ε^{-2}$, and a spatial location of order $ε^{-3/2}$. 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 $ε^{-2}$, this distribution function converges to the one of Tracy-Widom.