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October, 1980 On the Growth of One Dimensional Contact Processes
Richard Durrett
Ann. Probab. 8(5): 890-907 (October, 1980). DOI: 10.1214/aop/1176994619


In this paper we will study the number of particles alive at time $t$ in a one dimensional contact process $\xi^0_t$ which starts with one particle at 0 at time 0. In the case of a nearest neighbor interaction we will show that if $|\xi^0_t|$ is the number of particles and $r_t, l_t$ are the positions of the rightmost and leftmost particles (with $r_t = l_t = 0$ if $|\xi^0_t| = 0$) then there are constants $\gamma, \alpha$, and $\beta$ so that $|\xi^0_t|/t, r_t/t$, and $l_t/t$ converge in $L^1$ to $\gamma 1_\Lambda, \alpha 1_\Lambda$ and $\beta 1_\Lambda$ where $\Lambda = \{|\xi^0_t| > 0$ for all $t\}$. The constant $\gamma = \rho(\alpha - \beta)^+$ where $\rho$ is the density of the "upper invariant measure" $\xi^Z_\infty$.


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Richard Durrett. "On the Growth of One Dimensional Contact Processes." Ann. Probab. 8 (5) 890 - 907, October, 1980.


Published: October, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0457.60082
MathSciNet: MR586774
Digital Object Identifier: 10.1214/aop/1176994619

Primary: 60K35
Secondary: 60F15

Keywords: contact process , convergence theorem , coupling , Infinite particle system , Subadditive processes

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 5 • October, 1980
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