Shocks in the asymmetric exclusion process

In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density λ at -∞ and {telephone recorder} at +∞. Earlier results described the limiting behavior in all cases except for 0<λ<1/2, λ+{telephone recorder}=1. Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in 1975 [7]. The convergence proof is based on coupling and symmetry considerations.