Shocks in the asymmetric exclusion process

E. D. Andjel, M. D. Bramson, T. M. Liggett

Research output: Contribution to journalArticlepeer-review

52 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)231-247
Number of pages17
JournalProbability Theory and Related Fields
Issue number2
StatePublished - Jun 1988


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