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Published September 27, 2021 | Submitted
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Non-triviality of the phase transition for percolation on finite transitive graphs


We prove that if (G_n)_(n ≥ 1) = ((V_n,E_n))_(n ≥ 1) is a sequence of finite, vertex-transitive graphs with bounded degrees and |V_n|→∞ that is at least (1+ϵ)-dimensional for some ϵ > 0 in the sense that diam(G_n)=O(|V_n|^(1/(1+ϵ) as n → ∞ then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each 0<α<1 there exists p_c(α) < 1 such that for each p ≥ p_c(α), Bernoulli-p bond percolation on G_n has a cluster of size at least α|V_n| with probability tending to 1 as n → ∞. In fact, we prove more generally that there exists a universal constant a such that the same conclusion holds whenever diam(G_n) = 0(|V_n|/(log|V_n|α) as n → ∞. This verifies a conjecture of Benjamini up to the value of the constant a, which he suggested should be 1. We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits and prove that one may indeed take a = 1 when the graphs G_n are all Cayley graphs of Abelian groups. A key step in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a nontrivial uniqueness phase (i.e., to have p_u < 1) which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at 1 in the sense that for every k,n < ∞ there exists ϵ > 0 such that every infinite graph G of degree at most k whose vertex set has at most n orbits under Aut(G) either has p_c = 1 or p_c ≤ 1 - ϵ.

Additional Information

The first author was supported in part by ERC starting grant 804166 (SPRS) and thanks Gabor Pete for helpful discussions. The second author was partially supported by the Stokes Research Fellowship at Pembroke College, Cambridge. He is also grateful to Itai Benjamini and Ariel Yadin for introducing him to the problems considered in this paper, and to Romain Tessera for helpful conversations.

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