The Contact Process on a Finite Set

Abstract

In this paper we show that the phase transition in the contact process manifests itself in the behavior of large finite systems. To be precise, if we let $\sigma_N$ denote the time the process on $\{1, \cdots, N\}$ first hits $\varnothing$ starting from all sites occupied, then there is a critical value $\lambda_c$ so that (i) for $\lambda < \lambda_c$ there is a constant $\gamma(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty, \sigma_n /\log N \rightarrow 1/\gamma(\lambda)$ in probability and (ii) for $\lambda > \lambda_c$ there are constants $\alpha (\lambda), \beta(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty$, $P(\alpha(\lambda)/2 - \varepsilon \leq (\log \sigma_N)/N \leq \beta (\lambda) + \varepsilon) \rightarrow 1,$ for all $\varepsilon > 0$. Our results improve upon an earlier work of Griffeath but as the reader can see the second one still needs improvement. To help decide what should be true for the contact process we also consider the analogous problem for the biased voter model. For this process we can show $(\log \sigma_N)/N \rightarrow \alpha(\lambda) = \beta(\lambda)$ in probability, and it seems likely that the same result is true for the contact process.