## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 2 (1989), 433-443.

### Survival of Nearest-Particle Systems with Low Birth Rate

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 2 (1989), 433-443.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991409

**Digital Object Identifier**

doi:10.1214/aop/1176991409

**Mathematical Reviews number (MathSciNet)**

MR985372

**Zentralblatt MATH identifier**

0682.60091

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Nearest-particle system low birth rate survival

#### Citation

Bramson, Maury. Survival of Nearest-Particle Systems with Low Birth Rate. Ann. Probab. 17 (1989), no. 2, 433--443. doi:10.1214/aop/1176991409. https://projecteuclid.org/euclid.aop/1176991409