The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions $9,10,11,\dots$

Abstract

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.

58 pages, 8 figures. V2: Major revisions: Exposition improved and errors corrected. Proof of multicomponent indistinguishability has been removed and will reappear as a separate paper

References in corpus (4)

• Negative association in uniform forests and connected graphs
• On the easiest way to connect $k$ points in the Random Interlacements process
• Percolative properties of Brownian interlacements and its vacant set
• Connectivity properties of Branching Interlacements