Article,
Statistical mechanics of crabgrass
Ann. Probab., 17 (2): 444--481 (1989)This paper examines a version of the contact process with a large range. Particles die at rate 1, and a particle is created at an empty site $x$ at rate $łambda$ times the fraction of occupied sites in $y:||x-y||M$. This contact process is dominated by a branching random walk with death rate 1 and birth rate $łambda$, and it is shown that in many ways these two processes are very similar when $M$ is large. In particular, as $M\toınfty$, the critical value for the contact process converges to 1, which is the critical value for branching random walks. The authors obtain precise rates for this convergence, in every dimension, enabling them to describe the ``crossover'' from contact process to branching process behavior in terms of the survival probability of a process started from a single particle. The proofs of the main results use many estimates for branching random walks, further detailing the nature of this crossover behavior..
Abstract
In this article we consider the asymptotic behavior of the contact process when the range $M$ goes to $ınfty$. We show that if $łambda$ is the total birth rate from an isolated particle, then the critical value $łambda_c(M) 1$ as $M ınfty$. The rate of convergence depends upon the dimension: $łambda_c(M) - 1 M^-2/3$ in $d = 1, (M)/M^2$ in $d = 2$, and $M^-d$ in $d 3$.
