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July, 1988 The Contact Process on a Finite Set
Richard Durrett, Xiu-Fang Liu
Ann. Probab. 16(3): 1158-1173 (July, 1988). DOI: 10.1214/aop/1176991682


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.


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Richard Durrett. Xiu-Fang Liu. "The Contact Process on a Finite Set." Ann. Probab. 16 (3) 1158 - 1173, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0647.60105
MathSciNet: MR942760
Digital Object Identifier: 10.1214/aop/1176991682

Primary: 60K35

Keywords: Biased voter model , contact process

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 3 • July, 1988
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