The homogenous tree as an electric network

TL;DR

This paper explores the electrical network model of an infinite homogeneous tree, linking it to simple random walks and deriving key properties like voltages, currents, and hitting times through electric network theory.

Contribution

It provides a novel application of electric network theory to analyze random walks on homogeneous trees, simplifying proofs of their properties.

Findings

Explicit formulas for voltages and currents on the tree.

Expressions for Green function hitting times.

Probabilities of reaching sets before others.

Abstract

Let T be an infinite homogenous tree of homogeneity $q+1$. Attaching to each edge the conductance $1$, the tree will became an electric network. The reversible Markov chain associated to this network is the simple random walk on the homogenous tree. Using results regarding the equivalence between a reversible Markov chain and an electric network, we will express voltages, currents, the Green fuction hitting times, transitions number, probabilities of reaching a set before another, as functions of the distance on the homogenous tree. This connection enables us to give simpler proofs for the properties of the random walk under discussion.