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Exclusion processes in higher dimensions: stationary measures and convergence. (English) Zbl 1099.60067
The authors consider the exclusion process on \(Z^{d}\) with \(d>1\). Firstly, one gives a necessary and sufficient condition for a product measure to be stationary for the process. One applies then this condition to the case of a translation system on \(Z^{d}\). This allows to construct many examples of stationary product measures that are neither homogeneous nor reversible. Examples for a random walk on a homogeneous tree and on a rooted tree are given. Then the authors study stationary measures which are invariant under translations in all directions orthogonal to a fixed nonzero \(v\in Z^{d}\) and prove a number of convergence results for the measure of the exclusion process. Applications of hydrodynamical results are used to obtain explicit convergence results. Some open problems are also stated.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
Keywords:
hydrodynamics
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