## The Annals of Probability

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

Robin Pemantle

#### Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990223

Digital Object Identifier
doi:10.1214/aop/1176990223

Mathematical Reviews number (MathSciNet)
MR1127715

Zentralblatt MATH identifier
0758.60010

JSTOR