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Nearest particle systems: Results and open problems. (English) Zbl 0603.60094
Stochastic spatial processes, Proc. Conf., Heidelberg/Ger. 1984, Lect. Notes Math. 1212, 200-215 (1986).
[For the entire collection see Zbl 0593.00019.]
Nearest particle systems were introduced by F. Spitzer [Bull. Am. Math. Soc. 83, 880-890 (1977; Zbl 0372.60149)]. They are continuous time Markov processes on the set of configurations of particles located at different sites of \({\mathbb{Z}}\). The rate at which an occupied site becomes empty and vice versa depends on the rest of the configuration only through the distances to the nearest occupied sites to the right and left.
This paper gives a survey of the present stage of the theory with mention of some important open problems. It includes several examples. The results are classified according to reversible and general, finite and infinite systems. The main proofs are sketched. For detailed proofs the reader is referred to the author’s book ”Interacting particle systems.” (1985; Zbl 0559.60078).
This paper offers a brief and lucid introduction into this interesting type of interacting particle systems.
Reviewer: M.Mürmann

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J25 Continuous-time Markov processes on general state spaces