The Annals of Probability

Random Walks, Capacity and Percolation on Trees

Russell Lyons

Full-text: Open access

Abstract

A collection of several different probabilistic processes involving trees is shown to have an unexpected unity. This makes possible a fruitful interplay of these probabilistic processes. The processes are allowed to have arbitrary parameters and the trees are allowed to be arbitrary as well. Our work has five specific aims: First, an exact correspondence between random walks and percolation on trees is proved, extending and sharpening previous work of the author. This is achieved by establishing surprisingly close inequalities between the crossing probabilities of the two processes. Second, we give an equivalent formulation of these inequalities which uses a capacity with respect to a kernel defined by the percolation. This capacitary formulation extends and sharpens work of Fan on random interval coverings. Third, we show how this formulation also applies to generalize work of Evans on random labelling of trees. Fourth, the correspondence between random walks and percolation is used to decide whether certain random walks on random trees are transient or recurrent a.s. In particular, we resolve a conjecture of Griffeath on the necessity of the Nash-Williams criterion. Fifth, for this last purpose, we establish several new basic results on branching processes in varying environments.

Article information

Source
Ann. Probab. Volume 20, Number 4 (1992), 2043-2088.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176989540

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176989540

Mathematical Reviews number (MathSciNet)
MR1188053

Zentralblatt MATH identifier
0766.60091

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

Keywords
Trees random walks percolation branching processes varying environment Nash-Williams criterion random labelling random covering capacity

Citation

Lyons, Russell. Random Walks, Capacity and Percolation on Trees. Ann. Probab. 20 (1992), no. 4, 2043--2088. doi:10.1214/aop/1176989540. http://projecteuclid.org/euclid.aop/1176989540.


Export citation