## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60C05: Combinatorial probability

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

**Keywords**

Spanning tree spanning forest loop-erased random walk

#### Citation

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