Gap at 1 for the percolation threshold of Cayley graphs

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We prove that the set of possible values for the percolation threshold pc of Cayley graphs has a gap at 1 in the sense that there exists ϵ0 > 0 such that for every Cayley graph G one either has pc(G) = 1 or pc(G) ≤ 1 - ϵ0. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo & Yadin (Duke Math. J. 169 (2020) 3539 3563) to the existence of phase transition using the Gaussian free field, combined with the finitary version of Gromov s theorem on the structure of groups of polynomial growth of Breuillard, Green & Tao (Publ. Math. Inst. Hautes Études Sci. 116 (2012) 115 221).

Original languageEnglish
Pages (from-to)1248-1258
Number of pages11
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number3
Publication statusPublished - 31 Aug 2023

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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