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Hitting times for special patterns in the symmetric exclusion process on \(\mathbb{Z}^d\). (English) Zbl 1067.60096

A symmetric simple exclusion process on a cubic lattice \(Z^d\) is considered for indistinguishable particles. It is implemented by associating independent Poisson processes of intensity \(1\) with bonds of the lattice. At Poisson times the contents of corresponding adjacent sites are exchanged. For a specific local pattern in the state space, the problem of establishing sharp estimates for the hitting time of the pattern is addressed. The obtained exclusion process is a non-irreducible Markov process on an uncountable state which has an invariant measure and whose dynamical generator together with its dual are monotone. A general method is proposed to prove regularity of invariant measures. The paper belongs to a series of papers devoted to: regularity of quasi-stationary measures for simple exclusion [A. Asselah and P. A. Ferrari, Ann. Probab. 30, No. 4, 1913–1932 (2002; Zbl 1014.60089)] and existence of quasi-stationary measures for asymmetric attractive particle systems [A. Asselah and F. Castell, Ann. Appl. Probab. 13, No. 4, 1569–1590 (2003; Zbl 1079.60075)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
60J25 Continuous-time Markov processes on general state spaces
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References:

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