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October, 1991 Choosing a Spanning Tree for the Integer Lattice Uniformly
Robin Pemantle
Ann. Probab. 19(4): 1559-1574 (October, 1991). DOI: 10.1214/aop/1176990223


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.


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Robin Pemantle. "Choosing a Spanning Tree for the Integer Lattice Uniformly." Ann. Probab. 19 (4) 1559 - 1574, October, 1991.


Published: October, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0758.60010
MathSciNet: MR1127715
Digital Object Identifier: 10.1214/aop/1176990223

Primary: 60C05
Secondary: 60K35

Keywords: Loop-erased random walk , spanning forest , spanning tree

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • October, 1991
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