Uniform Spanning Tree

A uniform spanning tree of a finite connected graph is a spanning tree sampled with equal probability from all of the graph's spanning trees. For example, each of the three spanning trees of the cycle graph has probability . Sampling a uniform tree conditional on a random planar map differs from sampling a map-tree pair uniformly: the latter weights the marginal map by its number of spanning trees. Uniform spanning trees on finite exhaustions give the free uniform spanning forest and wired uniform spanning forest on infinite locally finite graphs, with the boundary convention determining the law.