The Annals of Probability

The Contact Process on a Finite Set

Richard Durrett and Xiu-Fang Liu

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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.

Article information

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991682

Digital Object Identifier
doi:10.1214/aop/1176991682

Mathematical Reviews number (MathSciNet)
MR942760

Zentralblatt MATH identifier
0647.60105

JSTOR
links.jstor.org

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

Keywords
Contact process biased voter model

Citation

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. https://projecteuclid.org/euclid.aop/1176991682


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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.