OCCUPATION STATISTICS OF CRITICAL BRANCHING RANDOM WALKS IN TWO OR HIGHER DIMENSIONS

• 成果类型: Article
• 署名作者: Lalley, Steven P.; Zheng, Xinghua
• 署名单位: University of Chicago; Hong Kong University of Science & Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP551
• 发表日期: 2011
• 页码: 327-368
• 关键词: behavior

摘要

Consider a critical nearest-neighbor branching random walk on the d-dimensional integer lattice initiated by a single particle at the origin. Let G(n) be the event that the branching random walk survives to generation n. We obtain the following limit theorems, conditional on the event G(n), for a variety of occupation statistics: ( I) Let V-n be the maximal number of particles at a single site at time n. If the offspring distribution has finite alpha th moment for some integer alpha >= 2, then, in dimensions 3 and higher, V-n = O-p(n(1/alpha)) If the offspring distribution has an exponentially decaying tail, then Vn = O-p(log n) in dimensions 3 and higher and V-n = O-p((log n)(2)) in dimension 2. Furthermore, if the offspring distribution is nondegenerate, then P (V-n >= delta log n vertical bar G(n)) -> 1 for some delta > 0. (2) Let M-n (j) be the number of multiplicity-j sites in the nth generation, that is, sites occupied by exactly j particles. In dimensions 3 and higher, the random variables M-n(j)/n converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a typical site (i.e., at the location of a randomly chosen particle of the nth generation) is of order O-p(log n) and the number of occupied sites is O-p(n/log n). We also show that, in dimension 2, there is particle clustering around a typical site.