The Annals of Probability

The Binary Contact Path Process

David Griffeath

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Abstract

We study some $\{0, 1, \cdots\}^{z^d}$ valued Markov interactions $\eta_t$ called contact path processes. These are similar to branching random walks, in that the normalized size process starting from a singleton is a martingale which converges to a limit $M_\infty$. In contrast to branching, however, $M_\infty$ depends on the spatial dynamics of the path process. The main result is an exact evaluation of the variance of $M_\infty$, achieved by means of the Feynman-Kac formula. The basic contact process of Harris may be viewed as a projection of $\eta_t$; as a corollary to the main result we obtain bounds on the contact process critical value $\lambda^{(d)}_c$ in dimension $d \geq 3$.

Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 692-705.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993514

Digital Object Identifier
doi:10.1214/aop/1176993514

Mathematical Reviews number (MathSciNet)
MR704556

Zentralblatt MATH identifier
0524.60096

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Contact processes interacting particle systems critical values phase transition Feynman-Kac formula

Citation

Griffeath, David. The Binary Contact Path Process. Ann. Probab. 11 (1983), no. 3, 692--705. doi:10.1214/aop/1176993514. https://projecteuclid.org/euclid.aop/1176993514


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