Cutoff and discrete product structure in ASEP

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We consider the asymmetric simple exclusion process (ASEP) on Z with an initial data such that in the large time particle density ρ(·) a discontinuity at the origin is created, where the value of ρ jumps from zero to one, but ρ(−ε),1 − ρ(ε) > 0 for any ε > 0. We consider the position of a particle x_M macroscopically located at the discontinuity, and show that its limit law has a cutoff under t^1/2 scaling. Inside the discontinuity region, we show that a discrete product limit law arises, which bounds from above the limiting fluctuations of x_M in the general ASEP, and equals them in the totally ASEP. Sending M → ∞, the discrete product limit law converges to F_GUE × F_GUE, which was previously observed at shocks in TASEP.

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