THE RANGE OF TREE-INDEXED RANDOM WALK IN LOW DIMENSIONS

• 成果类型: Article
• 署名作者: Le Gall, Jean-Francois; Lin, Shen
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP947
• 发表日期: 2015
• 页码: 2701-2728
• 关键词: branching random-walks

摘要

We study the range R-n, of a random walk on the d-dimensional lattice Z(d) indexed by a random tree with n vertices. Under the assumption that the random walk is centered and has finite fourth moments, we prove in dimension d <= 3 that n(-d/4)R(n) converges in distribution to the Lebesgue measure of the support of the integrated super-Brownian excursion (ISE). An auxiliary result shows that the suitably rescaled local times of the tree-indexed random walk converge in distribution to the density process of ISE. We obtain similar results for the range of critical branching random walk in Z(d), d <= 3. As an intermediate estimate, we get exact asymptotics for the probability that a critical branching random walk starting with a single particle at the origin hits a distant point. The results of the present article complement those derived in higher dimensions in our earlier work.