Wired uniform spanning forest

The wired uniform spanning forest of an infinite graph is the weak limit of uniform spanning trees on finite exhaustions whose exterior vertices are identified to one wired boundary vertex.

Wilson's algorithm rooted at infinity builds the wired uniform spanning forest on a transient graph by successively adding loop-erased random walks, run forever when they never hit the forest already constructed.

For a locally finite connected graph with positive electrical conductances, the number of trees in its wired uniform spanning forest is almost surely a deterministic constant, possibly infinite. A precise characterization uses independent random walks with the conductance transition probabilities. Let $\alpha (w_1,\dots ,w_k)$ be the probability that their full vertex ranges are pairwise disjoint. ThenThe right side depends only on the electrical network. The proof amplifies positive disjointness probability to probabilities arbitrarily close to one, then uses Wilson's algorithm rooted at infinity to obtain at least $k$ components. Conversely, $k$ distinct components force starting vertices admitting positive disjointness probability. This establishes the constant without treating component count as automatically invariant under finite edge changes.^[ref]