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.

The number of trees in a wired uniform spanning forest is almost surely constant. This follows from its tail triviality together with the fact that every component is infinite.