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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.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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