A central limit theorem for biased random walks on Galton-Watson trees

• 成果类型: Article
• 署名作者: Peres, Yuval; Zeitouni, Ofer
• 署名单位: University of California System; University of California Berkeley; Microsoft
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0077-y
• 发表日期: 2008
• 页码: 595-629
• 关键词: markov-chains; branching-processes; random environment; percolation; clusters

摘要

Let T be a rooted Galton-Watson tree with offspring distribution {p(k)} that has p(0) = 0, mean m = Sigma kp(k) > 1 and exponential tails. Consider the lambda-biased random walk {X-n}(n >= 0) on T; this is the nearest neighbor random walk which, when at a vertex v with d(v) offspring, moves closer to the root with probability lambda(lambda + d(v)), and moves to each of the offspring with probability 1/(lambda + d(v)). It is known that this walk has an a.s. constant speed v = lim(n)|X-n|/n ( where |X-n| is the distance of X-n from the root), with v > 0 for 0 < lambda < m and v = 0 for lambda >= m. For lambda <= m, we prove a quenched CLT for |X-n| - nv. ( For lambda > m the walk is positive recurrent, and there is no CLT.) The most interesting case by far is. = m, where the CLT has the following form: for almost every T, the ratio |X-[nt]|/root n converges in law as n -> infinity to a deterministic multiple of the absolute value of a Brownian motion. Our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle ( previously, such a measure was explicitly known only for lambda = 1) and the construction of appropriate harmonic coordinates.