Document Zbl 0462.60096

Infinite systems with locally interacting components. (English) Zbl 0462.60096

Ann. Probab. 9, 349-364 (1981).

This is the manuscript of a lecture addressed not only to specialists in the fleld of infinite interacting systems. Several classes of processes occur in it, though not all of them are treated equally extensively: the Glauber model (with a mentioning of the voter model), some recently at that time discovered processes [by the author and Th. I \( 18 g e t t \); for further reference see their joint paper: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrschalnliabkattatheon. Verw. Geb. 56, 443-468 (1981; Zbl. 444.60096)]. viz the smoothing and potlatch process and coupled random walks, finally generalizations of the voter model to the case of a contimuous state space at a single site. The main content of the paper, in the reviewers opinion, and the unifying principle connecting the different examples is the method of ”duality”, more generally the exploitation of concrete algebraic structures (In the case of the Glauber model). The results presented are theorems on the 1dent1fication of invariant measures and ergodic theorems. Due to the expository character of the lecture proves are only given for the Glauber model and, for the other models, in the case of a finite set of sites. To the reader interested in a more detalled study of smoothing and potlatch processes the above mentioned article by Spitzer and Liggett is recommended.

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