Dynamical percolation on general trees

• 成果类型: Article
• 署名作者: Khoshnevisan, Davar
• 署名单位: Utah System of Higher Education; University of Utah
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0061-6
• 发表日期: 2008
• 页码: 169-193
• 关键词: random-walks; sets; capacity; THEOREM

摘要

Haggstrom et al. (Ann Inst H Poincare Probab Stat 33(4):497-528, 1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree (Peres and Steif in Probab Theory Relat Fields 111(1):141-165, 1998), derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present paper is a necessary and sufficient condition for the existence of percolation, at some time t epsilon D in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also a formula for the Hausdorff dimension of the set of exceptional times of percolation.