The Annals of Probability

Choosing a Spanning Tree for the Integer Lattice Uniformly

Robin Pemantle

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Consider the nearest neighbor graph for the integer lattice $\mathbf{Z}^d$ 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 $\mathbf{Z}^d$. This is shown to be a tree if and only if $d \leq 4$. In this case, the tree has only one topological end, that is, there are no doubly infinite paths. When $d \geq 5$ the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.

Article information

Ann. Probab., Volume 19, Number 4 (1991), 1559-1574.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60C05: Combinatorial probability
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Spanning tree spanning forest loop-erased random walk


Pemantle, Robin. Choosing a Spanning Tree for the Integer Lattice Uniformly. Ann. Probab. 19 (1991), no. 4, 1559--1574. doi:10.1214/aop/1176990223.

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