Abstract
We prove that the set of possible values for the percolation threshold of Cayley graphs has a gap at 1 in the sense that there exists such that for every Cayley graph G one either has or . 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).
Nous prouvons que l’ensemble des valeurs possibles pour les seuils critiques de percolation de graphes de Cayley a une « lacune » en 1, dans le sens qu’il existe tel que pour tout graphe de Cayley G, on a soit , soit . La démonstration s’appuie sur la nouvelle approche de Duminil-Copin, Goswami, Raoufi, Severo & Yadin (Duke Math. J. 169 (2020) 3539–3563) pour prouver l’existence de la transition de phase en utilisant le champ libre gaussien, combinée avec la version finitaire du théorème de Gromov sur la structure des groupes à croissance polynomiale de Breuillard, Green & Tao (Publ. Math. Inst. Hautes Études Sci. 116 (2012) 115–221).
Funding Statement
This research was supported by the Swiss National Science Foundation and the NCCR SwissMAP.
The second author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 851565).
Acknowledgments
We would like to thank Hugo Duminil-Copin for inspiring discussions. We are also grateful to an anonymous referee for valuable comments on a prior version of this article.
Citation
Christoforos Panagiotis. Franco Severo. "Gap at 1 for the percolation threshold of Cayley graphs." Ann. Inst. H. Poincaré Probab. Statist. 59 (3) 1248 - 1258, August 2023. https://doi.org/10.1214/22-AIHP1286
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