Uniform spanning forests associated with biased random walks on Euclidean lattices

Abstract

The uniform spanning forest measure ($\mathsf{USF}$) 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 $\mathsf{USF}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\cite{PR1991} showed that on ${\mathbb{Z}}^d$, equipped with the unit conductance $c=1$, $\mathsf{USF}$ is concentrated on spanning trees if and only if $d\le 4$. In this work we study the $\mathsf{USF}$ associated with conductances induced by $\lambda$--biased random walk on ${\mathbb{Z}}^d$, $d\ge 2$, $0<\lambda <1$, i.e. conductances are set to be $c(e)={\lambda }^{-|e|}$, where $|e|$ is the graph distance of the edge $e$ from the origin. Our main result states that in this case $\mathsf{USF}$ 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 $d\ge 4$. Our method relies on the analysis of the spectral radius and the speed of the $\lambda$--biased random walk on ${\mathbb{Z}}^d$.

16 pages. Comments are welcome