Galton-Watson Trees with the Same Mean Have the Same Polar Sets

Robin Pemantle, Yuval Peres
1995 Annals of Probability  
Evans (1992) denes 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 ospring distribution has mean m and nite 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
more » ... space to trees and polar sets to capacity criteria, it also follows that certain random Cantor sets are capacity-equivalent to each other and to deterministic Cantor sets.
doi:10.1214/aop/1176988175 fatcat:n7vcjcqmbzh3bf7r2lfmu4rqoi