The Annals of Probability

The Contact Process on a Finite Set

Richard Durrett and Xiu-Fang Liu

Full-text: Open access


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.

Article information

Ann. Probab., Volume 16, Number 3 (1988), 1158-1173.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Contact process biased voter model


Durrett, Richard; Liu, Xiu-Fang. The Contact Process on a Finite Set. Ann. Probab. 16 (1988), no. 3, 1158--1173. doi:10.1214/aop/1176991682.

Export citation

See also

  • Part II: Richard Durrett, Roberto H. Schonmann. The Contact Process on a Finite Set. II. Ann. Probab., Volume 16, Number 4 (1988), 1570--1583.
  • Part III: Richard Durrett, Roberto H. Schonmann, Nelson I. Tanaka. The Contact Process on a Finite Set. III: The Critical Case. Ann. Probab., Volume 17, Number 4 (1989), 1303--1321.