Infinite weighted graphs with bounded resistance metric

TL;DR

This paper studies infinite weighted graphs with bounded resistance metrics, establishing boundary representations for harmonic functions, and showing that the metric completion is compact with implications for potential theory.

Contribution

It introduces explicit boundary representations for finite-energy harmonic functions on graphs with bounded resistance metrics and analyzes the properties of the metric completion.

Findings

Finite-energy harmonic functions form a Lipschitz algebra.

The metric completion of the graph is compact.

V is open in the completion.

Abstract

We consider infinite weighted graphs $G$, i.e., sets of vertices $V$, and edges $E$ assumed countable infinite. An assignment of weights is a positive symmetric function $c$ on $E$ (the edge-set), conductance. From this, one naturally defines a reversible Markov process, and a corresponding Laplace operator acting on functions on $V$, voltage distributions. The harmonic functions are of special importance. We establish explicit boundary representations for the harmonic functions on $G$ of finite energy. We compute a resistance metric $d$ from a given conductance function. (The resistance distance $d(x,y)$ between two vertices $x$ and $y$ is the voltage drop from $x$ to $y$, which is induced by the given assignment of resistors when 1 amp is inserted at the vertex $x$, and then extracted again at $y$.) We study the class of models where this resistance metric is bounded. We show that then the finite-energy functions form an algebra of $\frac{1}{2}$-Lipschitz-continuous and bounded functions on $V$, relative to the metric $d$. We further show that, in this case, the metric completion $M$ of $(V,d)$ is automatically compact, and that the vertex-set $V$ is open in $M$. We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for every function on $V$ of finite energy. We further compare $M$ to other compactifications; e.g., to certain path-space models.