INDISTINGUISHABILITY OF THE COMPONENTS OF RANDOM SPANNING FORESTS

• 成果类型: Article
• 署名作者: Timar, Adam
• 署名单位: Hungarian Academy of Sciences; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1225
• 发表日期: 2018
• 页码: 2221-2242
• 关键词: percolation; clusters

摘要

We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres and Schramm for Cayley graphs, which have been open up to now. Our methods apply to a more general class of percolations, those satisfying weak insertion tolerance, and work beyond Cayley graphs, in the more general setting of unimodular random graphs.