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 ε>0 such that for every Cayley graph 
one either has 
or 
. The proof builds on the new approach of Duminil-Copin, Goswami, Raoufi, Severo & Yadin 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.
In the first half of the talk, we will overview the problem of non-triviality of on transitive graphs, describe the main ideas of the recent work of Duminil-Copin et al, and give a high-level overview of the proof of our result. In the second half, we will go over our proof in more detail. Along the way, we will prove a new isoperimetric inequality for minimal Cayley graphs which allows us to obtain some strong estimates for the heat kernel of simple random walk.
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Based on
https://arxiv.org/abs/2112.10055