Z. Wahrscheinlichkeitstheorie verw. Gebiete 59, 535.~-552(1982) Zeitschrift fi)r Wahrscheinlichkeitstheorie und verwandte Gebiete ;9 Springer-Verlag 1982 Contact Processes in Several Dimensions Richard Durrett ~* and David Griffeath 2,, Mathematics Department, University of California, Los Angeles, CA90024, USA 2 Mathematics Department, University of Wisconsin, Van Vteck Hall, 480 Lincoln Drive, Madison, Wisconsin 53706, USA 1. Introduction This paper deals primarily with the basic contact processes introduced and studied by Harris [12-t4]. These are random evolutions on the state space S={all 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, {teS 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 proportional to the number of sites neighboring x where infection is present. The proportionality constant 2 is called the infection parameter. Contact processes are perhaps the simplest S-valued Markov pro- cesses which exhibit a "critical phenomenon": infection emanating from a single site dies out with probability 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 excep- tions: )~1) is not even determined to one decimal place (1.18<2~1)<2 are rigorous bounds), and very little is known about the behavior of the critical contact process (the case 2= 2~1~). If ({~(2)) is the d-dimensional basic contact process with parameter )o, starting at time 0 with infection everywhere on Z d, then for 2>2(~ e~ there is an * This work was done while the author was visiting Cornell, and partially supported by an NSF grant to that University ** This author was partially supported by NSF grant MCS78-01241 0044- 3 719/82/00 59/05 3 5/$03.60