The Contact Process on a Finite Set. II

Abstract

In this paper we complete the work started in part I. We show that if $\sigma_N$ is the time that the contact process on $\{1,\ldots, N\}$ first hits the empty set, then for $\lambda > \lambda_c$ (the critical value for the process on $Z$) there is a positive constant $\gamma(\lambda)$ so that $(\log \sigma_N)/N\rightarrow\gamma(\lambda)$ in probability as $N\rightarrow\infty$. We also give a new simple proof that $\sigma_N/E\sigma_N$ converges to a mean one exponential. The keys to the proof of the first result are a "planar graph duality" for the contact process and an observation of J. Chayes and L. Chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to $\infty$.