The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions $9,10,11,\ldots$.
Probability Theory and Related Fields(2018)
摘要
We prove that the uniform spanning forests of \(\mathbb {Z}^d\) and \(\mathbb {Z}^{\ell }\) have qualitatively different connectivity properties whenever \(\ell >d \ge 4\). 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.
更多查看译文
关键词
60D05, 60K35
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络