Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-13T07:25:09.133Z Has data issue: false hasContentIssue false

Ergodic theorems for graph interactions

Published online by Cambridge University Press:  01 July 2016

David Griffeath*
Affiliation:
Cornell University, New York

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
Copyright
Copyright © Applied Probability Trust 1975 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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, 7087.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, 325334.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[5] Holley, R. (1973) Unpublished.* Google Scholar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, 471481.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, 6472.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