The Annals of Probability

Statistical Mechanics of Crabgrass

M. Bramson, R. Durrett, and G. Swindle

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Abstract

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 $d \geq 3$.

Article information

Source
Ann. Probab., Volume 17, Number 2 (1989), 444-481.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991410

Digital Object Identifier
doi:10.1214/aop/1176991410

Mathematical Reviews number (MathSciNet)
MR985373

Zentralblatt MATH identifier
0682.60090

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Contact process renormalized bond construction branching processes

Citation

Bramson, M.; Durrett, R.; Swindle, G. Statistical Mechanics of Crabgrass. Ann. Probab. 17 (1989), no. 2, 444--481. doi:10.1214/aop/1176991410. https://projecteuclid.org/euclid.aop/1176991410


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