Choosing a Spanning Tree for the Integer Lattice Uniformly

Details Choosing a Spanning Tree for the Integer Lattice Uniformly Choosing a Spanning Tree for the Integer Lattice Uniformly

2004-04-02

arxiv.org/abs/math/0404043v1

Ann. Probab., 19, 1559 - 1574 (1991)

Mathematics

Probability

24 pages

Scientific paper

Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for Z^d. This is shown to be a tree if and only if d=<4. In this case, the tree has only one topological end, i.e. there are no doubly infinite paths. When d>=5 the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

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