RANDOM WALKS ON DISCRETE CYLINDERS WITH LARGE BASES AND RANDOM INTERLACEMENTS

• 成果类型: Article
• 署名作者: Windisch, David
• 署名单位: Weizmann Institute of Science
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP497
• 发表日期: 2010
• 页码: 841-895
• 关键词: percolation; graphs

摘要

Following the recent work of Sznitman [Probab. Theory Related Fields 145 (2009) 143-174], we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G(N) x Z, where G(N) is a large finite connected weighted graph, and relate it to the model of random interlacements on infinite transient weighted graphs. Under suitable assumptions, the set of points not visited by the random walk until a time of order vertical bar G(N)vertical bar(2) in a neighborhood of a point with Z-component of order vertical bar G(N)vertical bar converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point. The level of the random interlacement depends on the local time of a Brownian motion. The result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models. As examples of G(N), we treat the d-dimensional box of side length N, the Sierpinski graph of depth N and the d-ary tree of depth N, where d >= 2.