The Annals of Probability

Random Walks and Percolation on Trees

Russell Lyons

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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


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