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August, 1995 Survival of Discrete Time Growth Models, with Applications to Oriented Percolation
Thomas M. Liggett
Ann. Appl. Probab. 5(3): 613-636 (August, 1995). DOI: 10.1214/aoap/1177004698


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.


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Thomas M. Liggett. "Survival of Discrete Time Growth Models, with Applications to Oriented Percolation." Ann. Appl. Probab. 5 (3) 613 - 636, August, 1995.


Published: August, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0842.60090
MathSciNet: MR1359822
Digital Object Identifier: 10.1214/aoap/1177004698

Primary: 60K35

Keywords: contact process , critical values , percolation , survival

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.5 • No. 3 • August, 1995
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