How large a disc is covered by a random walk in n steps?

• 成果类型: Article
• 署名作者: Dembo, Amir; Peres, Yuval; Rosen, Jay
• 署名单位: Stanford University; Stanford University; City University of New York (CUNY) System; College of Staten Island (CUNY); University of California System; University of California Berkeley; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000854
• 发表日期: 2007
• 页码: 577-601
• 关键词: brownian-motion; 2 dimensions; points

摘要

We show that the largest disccovered by a simple random walk (SRW) on Z(2) after n steps has radius n(1/4+o(1)), thus resolving an open problem of Revesz [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. For any fixed , the largest disc completely covered at least times by the SRW also has radius n(1/4+o(l)). However, the largest disc completely covered by each of E independent simple random walks on Z(2) after n steps is only of radius n (1/(2+2 root)+o(l)). We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction alpha of the maximum number of visits to any site during the first n steps of the SRW on Z(2), is n((l-root alpha)/4+o(1)) .We also show that almost surely, for infinitely many values of n it takes about n (1/2+o(l)) steps after step n for the SRW to reach the first previously unvisited site (and the exponent 1/2 is sharp). This resolves a problem raised by Revesz.