The Contact Process on a Finite Set. III: The Critical Case

Abstract

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 contact process on $\mathbb{Z}, \sigma_N/N \rightarrow \infty$ and $\sigma_N/N^4 \rightarrow 0$ in probability as $N \rightarrow \infty$. The keys to the proof are a new renormalized bond construction and lower bounds for the fluctuations of the right edge. As a consequence of the result we get bounds on some critical exponents. We also study the analogous problem for bond percolation in $\{1,\ldots N\} \times \mathbb{Z}$ and investigate the limit distribution of $\sigma_N/E\sigma_N$.