The Annals of Probability

The Contact Process on a Finite Set. III: The Critical Case

Richard Durrett, Roberto H. Schonmann, and Nelson I. Tanaka

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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

Permanent link to this document
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
links.jstor.org

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

Keywords
Contact process critical exponent correlation length renormalization

Citation

Durrett, Richard; Schonmann, Roberto H.; Tanaka, Nelson I. The Contact Process on a Finite Set. III: The Critical Case. Ann. Probab. 17 (1989), no. 4, 1303--1321. doi:10.1214/aop/1176991156. https://projecteuclid.org/euclid.aop/1176991156


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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 II: Richard Durrett, Roberto H. Schonmann. The Contact Process on a Finite Set. II. Ann. Probab., Volume 16, Number 4 (1988), 1570--1583.