Uniform spanning forests associated with biased random walks on Euclidean lattices

The uniform spanning forest measure (𝖴𝖲𝖥) on a locally finite, infinite connected graph G 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 showed that on ℤ^d, 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 induced by λ–biased random walk on ℤ^d, d ≥ 2, 0 < λ < 1, i.e. conductances are set to be c(e) = λ^-|e|, where |e| is the graph distance of the edge e from the origin. 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^d 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^d.