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December, 1974 Contact Interactions on a Lattice
T. E. Harris
Ann. Probab. 2(6): 969-988 (December, 1974). DOI: 10.1214/aop/1176996493

Abstract

Let $\{\xi_t\}$ be a Markov process whose values are subsets of $Z_d$, the $d$-dimensional integers. Put $\xi_t(x) = 1$ if $x \in \xi_t$ and 0 otherwise. The transition intensity for a change in $\xi_t(x)$ depends on $\{\xi_t(y), y$ a neighbor of $x\}$. The chief concern is with "contact processes," where $\xi_t(x)$ can change from 0 to 1 only if $\xi_t(y) = 1$ for some $y$ neighboring $x$. Let $p_t(\xi) = \operatorname{Prob} \{\xi_t \neq \varnothing \mid \xi_0 = \xi\}$. Under appropriate conditions, $p_t$ is increasing, subadditive, or submodular in $\xi$. In the case of contact processes, conditions are giving implying that $p_\infty(\xi) = 0$ for all finite $\xi$, or that the contrary is true. In other cases conditions for ergodicity are given.

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T. E. Harris. "Contact Interactions on a Lattice." Ann. Probab. 2 (6) 969 - 988, December, 1974. https://doi.org/10.1214/aop/1176996493

Information

Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0334.60052
MathSciNet: MR356292
Digital Object Identifier: 10.1214/aop/1176996493

Subjects:
Primary: 60K35

Keywords: birth-death interaction , contact , ergodicity , interaction , subadditivity‎

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 6 • December, 1974
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