2018/05/04 by Zhan Shi, Vladas Sidoravičius, Shi, Zhan +7 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1805.01615
openalex publication_date 2018/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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~\citePR1991 showed that on ℤd, 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 ℤd, 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 2d 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 ℤd.