## 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

### Citations:

Zbl 1014.60089; Zbl 1079.60075
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### References:

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