Open Access
April, 1989 Statistical Mechanics of Crabgrass
M. Bramson, R. Durrett, G. Swindle
Ann. Probab. 17(2): 444-481 (April, 1989). DOI: 10.1214/aop/1176991410


In this article we consider the asymptotic behavior of the contact process when the range $M$ goes to $\infty$. We show that if $\lambda$ is the total birth rate from an isolated particle, then the critical value $\lambda_c(M) \rightarrow 1$ as $M \rightarrow \infty$. The rate of convergence depends upon the dimension: $\lambda_c(M) - 1 \approx M^{-2/3}$ in $d = 1, \approx (\log M)/M^2$ in $d = 2$, and $\approx M^{-d}$ in $d \geq 3$.


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M. Bramson. R. Durrett. G. Swindle. "Statistical Mechanics of Crabgrass." Ann. Probab. 17 (2) 444 - 481, April, 1989.


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

zbMATH: 0682.60090
MathSciNet: MR985373
Digital Object Identifier: 10.1214/aop/1176991410

Primary: 60K35
Secondary: 60J80

Keywords: branching processes , contact process , renormalized bond construction

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 2 • April, 1989
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