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April, 1989 Survival of Nearest-Particle Systems with Low Birth Rate
Maury Bramson
Ann. Probab. 17(2): 433-443 (April, 1989). DOI: 10.1214/aop/1176991409


Nearest-particle systems form a class of continuous-time interacting particle systems on $\mathbb{Z}$. The birth rate $\beta(l, r)$ at a given site depends on the distances $l$ and $r$ to the nearest occupied sites on the left and right; deaths occur at rate 1. Assume that $b(n) = \sum_{l + r = n} \beta(l, r), 2 \leq n < \infty, b(\infty) = \sum^\infty_{l =1} \beta(l, \infty) + \sum^\infty_{r=1} \beta(\infty, r)$, is constant. In Liggett [6] the question was posed whether for $b(n) \equiv 1 + \varepsilon, 2 \leq n \leq \infty$, with $0 < \varepsilon \leq 1$, there are such systems which survive for all $t$. Here, we answer affirmatively for all such $\varepsilon$ and construct a class of examples.


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Maury Bramson. "Survival of Nearest-Particle Systems with Low Birth Rate." Ann. Probab. 17 (2) 433 - 443, April, 1989.


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

zbMATH: 0682.60091
MathSciNet: MR985372
Digital Object Identifier: 10.1214/aop/1176991409

Primary: 60K35

Keywords: low birth rate , Nearest-particle system , survival

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 2 • April, 1989
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