Abstract
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 language | English |
|---|---|
| Pages (from-to) | 1248-1258 |
| Number of pages | 11 |
| Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |
| Volume | 59 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 31 Aug 2023 |
Funding
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).
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
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