Locality of percolation for Abelian Cayley graphs

• Sébastien Martineau ^[1] ; Vincent Tassion ^[2]
  1. [1] Weizmann Institute of Science

     Weizmann Institute of Science

     Israel

     
  2. [2] Swiss Federal Institute of Technology in Zurich

     Swiss Federal Institute of Technology in Zurich

     Zürich, Suiza

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 2, 2017, págs. 1247-1277
• Idioma: inglés
• DOI: 10.1214/15-AOP1086
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• Resumen

  • We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function pcpc defined on the set of Cayley graphs of Abelian groups of rank at least 22 is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.