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

The crossover regime for the weakly asymmetric simple exclusion process