Contact processes in several dimensions

Abstract

This paper deals primarily with the basic contact processes introduced and studied by Harris [12t4] . These are random evolutions on the state space S={a l l subsets of Z a} (Zd=the d-dimensional integer lattice) with extremely simple local dynamics. Namely, if we think of the contact process (~) as representing the spread of an infection, { t eS denoting the set of infected sites at time t, then x E it becomes healthy at exponential rate 1 while x ~ Z a {t is infected at a rate proport ional to the number of sites neighboring x where infection is present. The proportionali ty constant 2 is called the infection parameter. Contact processes are perhaps the simplest S-valued Markov processes which exhibit a phenomenon: infection emanating from a single site dies out with probabili ty one for small positive 2., but has positive probability of surviving for all time when ;, is large. There is a critical value 2~ d) where the ~ transition occurs. Principal objectives of analysis are the precise formulation of the critical phenomenon, and detailed description of the ergodic behavior both below and above )(a} ' c Our recent survey articles [10] and I-6] provide introductions to contact processes, and to the general field of interacting S-valued systems, respectively, We will make constant use of notation and techniques from the surveys, assuming that the reader is familiar with thern. The article [10] describes in some detail the current state of knowledge concerning one dimensional contact processes. The theory for d = 1 is now fairly complete with two major exceptions: )~1) is not even determined to one decimal place (1.18 2(~ e~ there is an