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.