Non-triviality of the phase transition for percolation on finite transitive graphs

We prove that if (Gₙ)n ≥1=((Vₙ,Eₙ))n≥ 1 is a sequence of finite, vertex-transitive graphs with bounded degrees and |Vₙ|→∞ that is at least (1+ε) -dimensional for some ε>0 in the sense that diam (Gₙ)=O(|Vₙ|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 pc(α)<1 such that for each p≥ pc(α) , Bernoulli- p bond percolation on Gₙ has a cluster of size at least α |Vₙ| 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 (Gₙ)=O({|Vₙ|}{(log |Vₙ|)ᵃ}) as n→∞. This verifies a conjecture of Benjamini (2001) 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. A key step in our argument is a direct proof of our result when the graphs Gₙ are all Cayley graphs of Abelian groups, in which case we show that one may indeed take a=1 . This result relies crucially on a new theorem of independent interest stating roughly that balls in arbitrary Abelian Cayley graphs can always be approximated by boxes in ᵈ with the standard generating set. Another key step is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin [Duke Math. J. 169 (2020)] from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a non-trivial uniqueness phase (i.e., to have pᵤ<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) has either p_c=1 or p_c≤ 1-ε .