# Gap at 1 for the percolation threshold of Cayley graphs

@inproceedings{Panagiotis2021GapA1, title={Gap at 1 for the percolation threshold of Cayley graphs}, author={Christoforos Panagiotis and Franco Severo}, year={2021} }

We prove that the set of possible values for the percolation threshold p c of Cayley graphs has a gap at 1 in the sense that there exists ε 0 > 0 such that for every Cayley graph G one either has p c ( G ) = 1 or p c ( G ) ≤ 1 − ε 0 . The proof builds on the new approach of Duminil-Copin, Goswami, Raouﬁ, Severo & Yadin ( Duke Math. J., 2020 ) 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…

## 2 Citations

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Abstract Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and…

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