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Ergodic theorems for graph interactions

Published online by Cambridge University Press:  01 July 2016

David Griffeath
David Griffeath*
Affiliation:
Cornell University, New York
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Abstract

A criterion for ergodicity of lattice interactions has been given by Dobrushin [2], and improved by Harris [3] and Holley [5]. In this paper we present a simplified derivation of these results, and obtain stronger conditions for interactions on the one- and two-dimensional integer lattices, and the 3–Bethe lattice.

Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 1 , March 1975 , pp. 179 - 194
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Choquet, G. (1969) Lectures on Analysis. W. A. Benjamin.Google Scholar
[2] Dobrushin, R. L. (1971) Markov processes with a large number of locally interacting components. Problemy Peredači. Informacii. 7, 2, 70–87.Google Scholar
[3] Harris, T. E. (1974) Contact interactions on a lattice. Ann. Probability 2, 6.Google Scholar
[4] Holley, R. (1972) An ergodic theorem for interacting systems with attractive interactions. Z. Wahrscheinlichkeitsth. 24, 325–334.Google Scholar
[5] Holley, R. (1973) Unpublished.* Google Scholar
[6] Liggett, T. M (1972). Existence theorems for infinite particle systems. Trans. Amer. Math. Soc. 165, 471–481.Google Scholar
[7] Spitzer, F. (1971) Random Fields and Interacting Particle Systems. Mathematical Association of America.Google Scholar
[8] Vasershtein, L. N. (1969) Markov processes on countable product spaces, describing large systems of automata. Problemy Peredači Informacii 3, 64–72.Google Scholar
* Supplementary reference:Google Scholar
[9] Holley, R. and Liggett, T. (1974) Ergodic theorems for weakly interacting infinite systems and the voting model. To appear.Google Scholar