The Component Graph of the Uniform Spanning Forest: Transitions in Dimensions

We prove that the uniform spanning forests of Z and Z have qualitatively different connectivity properties whenever ` > 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 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. Along the way, we prove an indistinguishability theorem for collections of finitely many distinct trees in the uniform spanning forest of Z.