ON THE CRITICAL PARAMETER OF INTERLACEMENT PERCOLATION IN HIGH DIMENSION

• 成果类型: Article
• 署名作者: Sznitman, Alain-Sol
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP545
• 发表日期: 2011
• 页码: 70-103
• 关键词: vacant set; heat kernels; graphs

摘要

The vacant set of random interlacements on Z(d), d >= 3, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039-2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831-858] that there is a nondegenerate critical value u(*) such that the vacant set at level u percolates when u < u(*) and does not percolate when u > u(*). We derive here an asymptotic upper bound on u(*), as d goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that u(*) is equivalent to log d for large d and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604-1627].