The Annals of Probability

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 $\sigma_N/E\sigma_N$.

Article information

Source
Ann. Probab., Volume 17, Number 4 (1989), 1303-1321.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991156

Digital Object Identifier
doi:10.1214/aop/1176991156

Mathematical Reviews number (MathSciNet)
MR1048928

Zentralblatt MATH identifier
0692.60085

JSTOR