## The Annals of Probability

- Ann. Probab.
- Volume 18, Number 3 (1990), 931-958.

### Random Walks and Percolation on Trees

#### Abstract

There is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree. It equals the exponential of the Hausdorff dimension of the boundary in an appropriate metric. Its importance for probabilistic processes on a tree is shown in several ways, including random walk and percolation, where it provides points of phase transition.

#### Article information

**Source**

Ann. Probab., Volume 18, Number 3 (1990), 931-958.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990730

**Digital Object Identifier**

doi:10.1214/aop/1176990730

**Mathematical Reviews number (MathSciNet)**

MR1062053

**Zentralblatt MATH identifier**

0714.60089

**JSTOR**

links.jstor.org

**Subjects**

Primary: 05C05: Trees

Secondary: 60J15 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 82A43

**Keywords**

Trees random walks percolation random networks Hausdorff dimension random fractals branching processes

#### Citation

Lyons, Russell. Random Walks and Percolation on Trees. Ann. Probab. 18 (1990), no. 3, 931--958. doi:10.1214/aop/1176990730. https://projecteuclid.org/euclid.aop/1176990730