Asymptotic density in a coalescing random walk model

• 成果类型: Article
• 署名作者: Van den Berg, J; Kesten, H
• 署名单位: Cornell University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 303-352
• 关键词: annihilating random-walks

摘要

We consider a system of particles, each of which performs a continuous time random walk on Z(d). The particles interact only at times when a particle jumps to a site at which there are a number of other particles present. If there are j particles present, then the particle which just jumped is removed from the system with probability p(j). We show that if p(j) is increasing in j and if the dimension d is at least 6 and if we start with one particle at each site of Z(d), then p(t) := P{there is at least one particle at the origin at time t} similar to C(d)/t. The constant C(d) is explicitly identified. We think the result holds for every dimension d greater than or equal to 3 and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption d greater than or equal to 6. The proof is based on a justification of a certain mean field approximation for dp(t)/dt. The method seems applicable to many more models of coalescing and annihilating particles.