Infinite weighted graphs with bounded resistance metric

• Palle Jorgensen ^[1] ; Feng Tian ^[2]
  1. [1] University of Iowa

     University of Iowa

     City of Iowa City, Estados Unidos

     
  2. [2] Hampton University

     Hampton University

     Estados Unidos

     
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 123, Nº 1, 2018, págs. 5-38
• Idioma: inglés
• DOI: 10.7146/math.scand.a-106208
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.