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

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.