Geometry of the random walk range conditioned on survival among Bernoulli obstacles

• 成果类型: Article
• 署名作者: Ding, Jian; Fukushima, Ryoki; Sun, Rongfeng; Xu, Changji
• 署名单位: University of Pennsylvania; Kyoto University; National University of Singapore; University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00943-z
• 发表日期: 2020
• 页码: 91-145
• 关键词: brownian-motion; localization; confinement; asymptotics

摘要

We consider a discrete time simple symmetric random walk among Bernoulli obstacles on Z(d), d >= 2, where the walk is killed when it hits an obstacle. It is known that conditioned on survival up to time N, the random walk range is asymptotically contained in a ball of radius (sic)N = CN1/(d+ 2) for any d = 2. For d = 2, it is also known that the range asymptotically contains a ball of radius (1-) N for any > 0, while the case d = 3 remains open. We complete the picture by showing that for any d = 2, the random walk range asymptotically contains a ball of radius N - N for some . (0, 1). Furthermore, we show that its boundary is of size at most d-1 N (log N) a for some a > 0.