GALTON-WATSON TREES WITH THE SAME MEAN HAVE THE SAME POLAR SETS

• 成果类型: Article
• 署名作者: PEMANTLE, R; PERES, Y
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988175
• 发表日期: 1995
• 页码: 1102-1124
• 关键词: random-walks; percolation; capacity

摘要

Evans defined a notion of what it means for a set B to be polar for a process indexed by a tree. The main result herein is that a tree picked from a Galton-Watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies that deterministic and nondeterministic trees behave identically in a variety of probability models. Mapping subsets of Euclidean space to trees and polar sets to capacity criteria, it follows that certain random Canter sets are capacity-equivalent to each other and to deterministic Canter sets. An extension to branching processes in varying environment is also obtained.