Non-triviality of the phase transition for percolation on finite transitive graphs
Tom Hutchcroft
CalTech, Pasadena, USAMatthew Tointon
University of Bristol, Bristol, UK
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Abstract
We prove that if is a sequence of finite, vertex-transitive graphs with bounded degrees and that is at least -dimensional for some in the sense that
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 there exists such that for each , Bernoulli- bond percolation on has a cluster of size at least with probability tending to as . In fact, we prove more generally that there exists a universal constant such that the same conclusion holds whenever
This verifies a conjecture of Benjamini (2001) up to the value of the constant , which he suggested should be . 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 are all Cayley graphs of Abelian groups, in which case we show that one may indeed take . 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 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 ), 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 in the sense that for every there exists such that every infinite graph of degree at most whose vertex set has at most orbits under has either or .
Cite this article
Tom Hutchcroft, Matthew Tointon, Non-triviality of the phase transition for percolation on finite transitive graphs. J. Eur. Math. Soc. (2024), published online first
DOI 10.4171/JEMS/1453