The component graph of the uniform spanning forest: transitions in dimensions 9, 10, 11, ...

• 成果类型: Article
• 署名作者: Hutchcroft, Tom; Peres, Yuval
• 署名单位: University of Cambridge; Microsoft
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0884-3
• 发表日期: 2019
• 页码: 141-208
• 关键词: parabolic harnack inequality; random interlacements; markov-chains; indistinguishability

摘要

We prove that the uniform spanning forests of Z(d) and Z(l) have qualitatively different connectivity properties whenever l > d >= 4. In particular, we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex, which we call the component graph. We introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8. To separate dimensions 5, 6, 7, and 8, we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph. Our result sharpens a theorem of Benjamini, Kesten, Peres, and Schramm, who proved that the diameter of the component graph increases by one every time the dimension increases by four.