The Annals of Probability

The Contact Process on a Finite Set. II

Richard Durrett and Roberto H. Schonmann

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Abstract

In this paper we complete the work started in part I. 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 process on $Z$) there is a positive constant $\gamma(\lambda)$ so that $(\log \sigma_N)/N\rightarrow\gamma(\lambda)$ in probability as $N\rightarrow\infty$. We also give a new simple proof that $\sigma_N/E\sigma_N$ converges to a mean one exponential. The keys to the proof of the first result are a "planar graph duality" for the contact process and an observation of J. Chayes and L. Chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to $\infty$.

Article information

Source
Ann. Probab., Volume 16, Number 4 (1988), 1570-1583.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991584

Digital Object Identifier
doi:10.1214/aop/1176991584

Mathematical Reviews number (MathSciNet)
MR958203

Zentralblatt MATH identifier
0664.60106

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; Schonmann, Roberto H. The Contact Process on a Finite Set. II. Ann. Probab. 16 (1988), no. 4, 1570--1583. doi:10.1214/aop/1176991584. https://projecteuclid.org/euclid.aop/1176991584


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See also

  • Part I: Richard Durrett, Xiu-Fang Liu. The Contact Process on a Finite Set. Ann. Probab., Volume 16, Number 3 (1988), 1158--1173.
  • 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.