Random walks and percolation on trees. (English) Zbl 0714.60089

Lyons, Russell

Ann. Probab. 18, No. 3, 931-958 (1990).

H. Furstenberg [Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 41-59 (1970; Zbl 0208.322)] defined the branching number of a countable locally finite tree as the exponential of the Hausdorff dimension of its boundary in a suitable metric. This paper investigates several probabilistic aspects of the branching number concerning random walks and percolation on trees. Consider a random walk on a tree with a root, where at each vertex the probability of going along the edge towards the root is \(\lambda\) times the corresponding probability for every edge leading away from the root. Then the branching number characterizes the transition from transience to recurrence. In the percolation problem the edges are removed independently with probability 1-p. The critical percolation probability is the reciprocal of the branching number. This result is applied to supercritical branching and multitype branching processes and random resistive and capacitative networks. The basic tool of the whole paper is the correspondence to electrical networks on trees, in particular the study of flows and their energy.

Reviewer: M.Mürmann

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G50	Sums of independent random variables; random walks
05C80	Random graphs (graph-theoretic aspects)
05C05	Trees
60D05	Geometric probability and stochastic geometry
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
82B43	Percolation

Keywords:

networks; Hausdorff dimension; percolation on trees; transition from transience to recurrence; multitype branching processes; random resistive and capacitative networks

Citations:

Zbl 0208.322