Counting planar random walk holes

• 成果类型: Article
• 署名作者: Benes, Christian
• 署名单位: Tufts University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117907000000204
• 发表日期: 2008
• 页码: 91-126
• 关键词: brownian intersection exponents; VALUES

摘要

We study two variants of the notion of holes formed by planar simple random walk of time duration 2n and the areas associated with them. We prove in both cases that the number of holes of area greater than A(n), where (A (n)) is an increasing sequence, is, up to a logarithmic correction term, asymptotic to n center dot A(n)(-1) for a range of large holes, thus confirming an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to n. We also discuss the different exponent of 5/3 observed by Mandelbrot for small holes.