Explosion and linear transit times in infinite trees

• 成果类型: Article
• 署名作者: Amini, Omid; Devroye, Luc; Griffiths, Simon; Olver, Neil
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); McGill University; University of Oxford; Vrije Universiteit Amsterdam; Centrum Wiskunde & Informatica (CWI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0683-z
• 发表日期: 2017
• 页码: 325-347
• 关键词: minimal displacement; CONVERGENCE

摘要

Let T be an infinite rooted tree with weights assigned to its edges. Denote by the minimum weight of a path from the root to a node of the nth generation. We consider the possible behaviour of with focus on the two following cases: we say T is explosive if lim(n ->infinity) m(n)(T) < infinity, and say that T exhibits linear growth if lim inf(n -> 8) m(n)(T)/n > 0. We consider a class of infinite randomly weighted trees related to the Poisson-weighted infinite tree, and determine precisely which trees in this class have linear growth almost surely. We then apply this characterization to obtain new results concerning the event of explosion in infinite randomly weighted spherically-symmetric trees, answering a question of Pemantle and Peres (Ann Probab 22(1), 180-194, 1994). As a further application, we consider the random real tree generated by attaching sticks of deterministic decreasing lengths, and determine for which sequences of lengths the tree has finite height almost surely.