ON THE RANGE OF A RANDOM WALK IN A TORUS AND RANDOM INTERLACEMENTS

• 成果类型: Article
• 署名作者: Procaccia, Eviatar B.; Shellef, Eric
• 署名单位: University of California System; University of California Los Angeles; Weizmann Institute of Science
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP924
• 发表日期: 2014
• 页码: 1590-1634
• 关键词: vacant set; percolation

摘要

Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices visited by the walk. Distance and mixing bounds for the typical range are proven that are a k-iterated log factor from those on the full torus for arbitrary k. The proof uses hierarchical renormalization and techniques that can possibly be applied to other random processes in the Euclidean lattice. We use the same technique to bound the heat kernel of a random walk on random interlacements.