BIASED RANDOM WALKS ON GALTON-WATSON TREES WITH LEAVES

• 成果类型: Article
• 署名作者: Ben Arous, Gerard; Fribergh, Alexander; Gantert, Nina; Hammond, Alan
• 署名单位: New York University; University of Munster; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP620
• 发表日期: 2012
• 页码: 280-338
• 关键词: transient random-walks; random environment; percolation; LAW

摘要

We consider a biased random walk X-n on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant gamma = gamma(beta) is an element of (0, 1), depending on the bias beta, such that vertical bar X-n vertical bar is of order n(gamma). Denoting Delta(n) the hitting time of level n, we prove that Delta(n)/n(1/gamma) is tight. Moreover, we show that Delta(n)/n(1/gamma) does not converge in law (at least for large values of beta). We prove that along the sequences n(lambda)(k) = left perpendicular lambda beta(gamma k)right perpendicular, Delta(n)/n(1/gamma) converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.