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Shocks in the asymmetric exclusion process. (English) Zbl 0632.60107
We consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density \(\lambda\) at -\(\infty\) and \(\rho\) at \(+\infty\). Earlier results described the limiting behavior in all cases except for \(0<\lambda <1/2\), \(\lambda +\rho =1.\)
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 ‘Interacting particle systems.’ (1985; Zbl 0559.60078). The convergence proof is based on coupling and symmetric considerations.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F15 Strong limit theorems
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