The free uniform spanning forest on an infinite connected locally finite graph is the weak limit of uniform spanning trees on finite connected subgraphs exhausting the graph, retaining free boundary conditions. Vertices on the boundary are not identified with one another. The result is a probability law on spanning forests of the infinite graph (Benjamini et al. 2001, §5). Wiring the boundary before taking uniform spanning trees instead defines the wired uniform spanning forest, a different law in general.
Free Uniform Spanning Forest
See also
Factor of IID, Spanning Forest, Spanning Tree, Weak ConvergenceExplore with Wolfram|Alpha
References
Benjamini, I.; Lyons, R.; Peres, Y.; and Schramm, O. "Uniform Spanning Forests." Ann. Prob. 29, 1-65, 2001. https://doi.org/10.1214/aop/1008956321.Cite this as:
Weisstein, Eric W. "Free Uniform Spanning Forest." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FreeUniformSpanningForest.html