Note on uniformly transient graphs

• Matthias Keller ^[1] ; Daniel Lenz ^[1] ; Marcel Schmidt ^[1] ; Radosław K. Wojciechowski ^[2]
  1. [1] Friedrich Schiller University Jena

     Friedrich Schiller University Jena

     Kreisfreie Stadt Jena, Alemania

     
  2. [2] City University of New York

     City University of New York

     Estados Unidos

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 33, Nº 3, 2017 (Ejemplar dedicado a: Yves Meyer), págs. 831-860
• Idioma: inglés
• DOI: 10.4171/RMI/957
• Texto completo no disponible (Saber más ...)
• Resumen

  • We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have ℓp independent pure point spectra (for 1≤p≤∞).

    Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compactification and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.