## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 1 (1995), 87-104.

### Percolation and Minimal Spanning Forests in Infinite Graphs

#### Abstract

The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value $f(b)$ attached to each bond $b$. Of particular interest are stationary random graphs; examples include a lattice with iid uniform values $f(b)$ and the Voronoi or complete graph on the sites of a Poisson process, with $f(b)$ the length of $b$. The corresponding percolation models are Bernoulli bond percolation and the "lily pad" model of continuum percolation, respectively. It is shown that under a mild "simultaneous uniqueness" hypothesis, with at most one exception, each tree in the forest has one topological end, that is, has no doubly infinite paths. If there is a tree in the forest, necessarily unique, with two topological ends, it must contain all sites of an infinite cluster at the critical point in the corresponding percolation model. Trees with zero, or three or more, topological ends are not possible. Applications to invasion percolation are given. If all trees are one-ended, there is a unique optimal (locally minimax for $f$) path to infinity from each site.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 1 (1995), 87-104.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988378

**Digital Object Identifier**

doi:10.1214/aop/1176988378

**Mathematical Reviews number (MathSciNet)**

MR1330762

**Zentralblatt MATH identifier**

0827.60079

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 82B43: Percolation [See also 60K35] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C80: Random graphs [See also 60B20]

**Keywords**

Minimal spanning tree percolation continuum percolation invasion percolation

#### Citation

Alexander, Kenneth S. Percolation and Minimal Spanning Forests in Infinite Graphs. Ann. Probab. 23 (1995), no. 1, 87--104. doi:10.1214/aop/1176988378. https://projecteuclid.org/euclid.aop/1176988378