Infinite weighted graphs with bounded resistance metric

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.