FROZEN PERCOLATION ON THE BINARY TREE IS NONENDOGENOUS

• 成果类型: Article
• 署名作者: Rath, Balazs; Swart, Jan M.; Terpai, Tamas
• 署名单位: Budapest University of Technology & Economics; MTA-BME Stochastics Research Group; Czech Academy of Sciences; Institute of Information Theory & Automation of the Czech Academy of Sciences; Eotvos Lorand University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1507
• 发表日期: 2021
• 页码: 2272-2316
• 关键词: clusters

摘要

In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time an edge opens provided neither of its end vertices is part of an infinite open cluster; in the opposite case it freezes. Aldous (Math. Proc. Cambridge Philos. Soc. 128 (2000) 465477) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (Ann. Appl. Probab. 15 (2005) 1047-1110), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton-Watson tree that has nice scale invariant properties.