Resumen de Infinite weighted graphs with bounded resistance metric
We consider infinite weighted graphsG, i.e., sets of verticesV, and edgesEassumed countablyinfinite. An assignment of weights is a positive symmetric functionconE(the edge-set), conduct-ance. From this, one naturally defines a reversible Markov process, and a corresponding Laplaceoperator acting on functions onV, voltage distributions. The harmonic functions are of specialimportance. We establish explicit boundary representations for the harmonic functions onGoffinite energy.We compute a resistance metricdfrom a given conductance function. (The resistance distanced(x, y)between two verticesxandyis the voltage drop fromxtoy, which is induced by thegiven assignment of resistors when 1 amp is inserted at the vertexx, and then extracted again aty.)We study the class of models where this resistance metric is bounded. We show that then thefinite-energy functions form an algebra of 1/2-Lipschitz-continuous and bounded functions onV, relative to the metricd. We further show that, in this case, the metric completionMof(V , d)is automatically compact, and that the vertex-setVis open inM. We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for everyfunction onVof finite energy. We further compareMto other compactifications; e.g., to certain path-space models.
