The Annals of Applied Probability

Survival of Discrete Time Growth Models, with Applications to Oriented Percolation

Thomas M. Liggett

Full-text: Open access

Abstract

We prove survival for a class of discrete time Markov processes whose states are finite sets of integers. As applications, we obtain upper bounds for the critical values of various two-dimensional oriented percolation models. The technique of proof is based generally on that used by Holley and Liggett to prove survival of the one-dimensional basic contact process. However, the fact that our processes evolve in discrete time requires that we make substantial changes in the way this technique is used. When applied to oriented percolation on the two-dimensional square lattice, our result gives the following bounds: $p_c \leq 2/3$ for bond percolation and $p_c \leq 3/4$ for site percolation.

Article information

Source
Ann. Appl. Probab., Volume 5, Number 3 (1995), 613-636.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004698

Digital Object Identifier
doi:10.1214/aoap/1177004698

Mathematical Reviews number (MathSciNet)
MR1359822

Zentralblatt MATH identifier
0842.60090

JSTOR
links.jstor.org

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

Keywords
Survival contact process percolation critical values

Citation

Liggett, Thomas M. Survival of Discrete Time Growth Models, with Applications to Oriented Percolation. Ann. Appl. Probab. 5 (1995), no. 3, 613--636. doi:10.1214/aoap/1177004698. https://projecteuclid.org/euclid.aoap/1177004698


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