THE SLOW REGIME OF RANDOMLY BIASED WALKS ON TREES

• 成果类型: Article
• 署名作者: Hu, Yueyun; Shi, Zhan
• 署名单位: Universite Paris 13
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1064
• 发表日期: 2016
• 页码: 3893-3933
• 关键词: transient random-walks; random environment; branching-processes; percolation; BEHAVIOR; CONVERGENCE; martingale

摘要

We are interested in the randomly biased random walk on the supercritical Galton-Watson tree. Our attention is focused on a slow regime when the biased random walk (X-n) is null recurrent, making a maximal displacement of order of magnitude (log n)(3) in the first n steps. We study the localization problem of X-n and prove that the quenched law of X-n can be approximated by a certain invariant probability depending on n and the random environment. As a consequence, we establish that upon the survival of the system, vertical bar X-n vertical bar/(log n)(2) converges in law to some non-degenerate limit on (0, infinity) whose law is explicitly computed.