Gumbel fluctuations for cover times in the discrete torus

• 成果类型: Article
• 署名作者: Belius, David
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0467-7
• 发表日期: 2013
• 页码: 635-689
• 关键词: random-walks; vacant set; percolation

摘要

This work proves that the fluctuations of the cover time of simple random walk in the discrete torus of dimension at least three with large side-length are governed by the Gumbel extreme value distribution. This result was conjectured for example in Aldous and Fill (Reversible Markov chains and random walks on graphs, in preparation). We also derive some corollaries which qualitatively describe how covering happens. In addition, we develop a new and stronger coupling of the model of random interlacements, introduced by Sznitman (Ann Math (2) 171(3):2039-2087, 2010), and random walk in the torus. This coupling is used to prove the cover time result and is also of independent interest.