The Annals of Probability

Exclusion processes in higher dimensions: Stationary measures and convergence

M. Bramson and T. M. Liggett

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There has been significant progress recently in our understanding of the stationary measures of the exclusion process on Z. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on Zd and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on Zd. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as t→∞ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.

Article information

Ann. Probab., Volume 33, Number 6 (2005), 2255-2313.

First available in Project Euclid: 7 December 2005

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Exclusion process stationary measures hydrodynamics


Bramson, M.; Liggett, T. M. Exclusion processes in higher dimensions: Stationary measures and convergence. Ann. Probab. 33 (2005), no. 6, 2255--2313. doi:10.1214/009117905000000341.

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