Indistinguishability of trees in uniform spanning forests

• 成果类型: Article
• 署名作者: Hutchcroft, Tom; Nachmias, Asaf
• 署名单位: University of British Columbia; Tel Aviv University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0707-3
• 发表日期: 2017
• 页码: 113-152
• 关键词: abelian sandpile model; invariant percolation; limit; graphs

摘要

We prove that in both the free and the wired uniform spanning forest (FUSF and WUSF) of any unimodular random rooted network (in particular, of any Cayley graph), it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini et al. (Ann Probab 29(1):1-65, 2001). We also answer positively two additional questions of Benjamini et al. (Ann Probab 29(1):1-65, 2001) under the assumption of unimodularity. We prove that on any unimodular random rooted network, the FUSF is either connected or has infinitely many connected components almost surely, and, if the FUSF and WUSF are distinct, then every component of the FUSF is transient and infinitely-ended almost surely. All of these results are new even for Cayley graphs.