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October, 1989 The Contact Process on a Finite Set. III: The Critical Case
Richard Durrett, Roberto H. Schonmann, Nelson I. Tanaka
Ann. Probab. 17(4): 1303-1321 (October, 1989). DOI: 10.1214/aop/1176991156


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 $\sigma_N/E\sigma_N$.


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Richard Durrett. Roberto H. Schonmann. Nelson I. Tanaka. "The Contact Process on a Finite Set. III: The Critical Case." Ann. Probab. 17 (4) 1303 - 1321, October, 1989.


Published: October, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0692.60085
MathSciNet: MR1048928
Digital Object Identifier: 10.1214/aop/1176991156

Primary: 60K35

Keywords: contact process , correlation length , Critical exponent , renormalization

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 4 • October, 1989
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