Uniformity of the late points of random walk on Znd for d ≥ 3

• 成果类型: Article
• 署名作者: Miller, Jason; Sousi, Perla
• 署名单位: University of Cambridge
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0697-1
• 发表日期: 2017
• 页码: 1001-1056
• 关键词: brownian-motion; cover times; set

摘要

Suppose that X is a simple random walk on for and, for each t, we let consist of those which have not been visited by X by time t. Let be the expected amount of time that it takes for X to visit every site of . We show that there exists and a time as such that the following is true. For (resp. ), the total variation distance between the law of and the law of i.i.d. Bernoulli random variables indexed by with success probability tends to 0 (resp. 1) as . Let be the first time t that . We also show that the total variation distance between the law of and the law of a uniformly chosen set from with size tends to 0 (resp. 1) for (resp. ) as n -> infinity 3.