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Ergodic theorem for generalized long-range exclusion processes with positive recurrent transition probabilities. (English) Zbl 0657.60126
Let S be a set of sites, finite or countable. Consider a configuration of particles in these sites, with at most m particles at each site, m bigger than a fixed number. The k particles at site \(s\in S\) wait for an exponential period of time of parameter g(k), independent of all other sites and then a particle begins to move according to a Markov chain rule U on the sites until it finds a site which has less than m particles. This search is instantaneous and is not time consuming.
The author proved that there is a Markov process defined on the set of configurations which is a model for the above heuristic description. In the case that the associated Markov chain U is positive recurrent, then for each \(0\leq n<\infty\) there is an ergodic stationary measure for configuration with a total of n particles. And for an infinite number of particles, the only invariant measure is with all sites occupied with m particles. Also any probability measure on the configurations will converge weakly to a mixture of the above mentioned invariant measures.
Reviewer: M.Smorodinsky

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60J25 Continuous-time Markov processes on general state spaces
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References:
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