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Regularity of quasi-stationary measures for simple exclusion in dimension \(d\geq 5\). (English) Zbl 1014.60089

This paper deals with the standard, symmetric, simple exclusion process on \(d\)-dimensional lattice. It is well known that the process has a family of stationary distributions. The authors prove that when \(d\geq 5\), the existence and regularity of the density for the limiting measure when time goes to infinity, before the origin is occupied, with respect to each stationary distribution. As usual, the convergence rate is described by the first Dirichlet eigenvalue. In the lower dimensional cases, the behavior is completely different and was proven before. In particular, in dimension one or two, the limiting measure is the simple one: dies out everywhere. In dimension three or four, it is conjectured that the limiting measure would be singular with respect to the stationary one. A hard part of the proof is a generalization of Andjel’s inequality for correlations.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
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References:

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