Uniform spanning forests on biased Euclidean lattices

Abstract

The uniform spanning forest measure (USF) on a locally finite, infinite connected graph with conductance c is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding USF is not necessarily concentrated on the set of spanning trees. Pemantle [20] showed that on Z, equipped with the unit conductance c = 1, USF is concentrated on spanning trees if and only if d ≤ 4. In this work we study the USF associated with conductances c(e) = λ−|e|, where |e| is the graph distance of the edge e from the origin, and λ ∈ (0, 1) is a fixed parameter. Our main result states that in this case USF consists of finitely many trees if and only if d = 2 or 3. More precisely, we prove that the uniform spanning forest has 2 trees if d = 2 or 3, and infinitely many trees if d ≥ 4. Our method relies on the analysis of the spectral radius and the speed of the λ-biased random walk on Z. AMS 2010 subject classifications. Primary 60J10, 60G50, 05C81; secondary 60C05, 05C63, 05C80.