Description
We prove that Bernoulli percolation on bounded degree graphs with
isoperimetric dimension d>4 undergoes a non-trivial phase transition
(in the sense that pc<1). As a corollary, we obtain that the critical
point of Bernoulli percolation on infinite quasi-transitive graphs (in
particular, Cayley graphs) with super-linear growth is strictly
smaller than 1, thus answering a conjecture of Benjamini and Schramm.
The proof relies on a new technique consisting in expressing certain
functionals of the Gaussian Free Field (GFF) in terms of connectivity
probabilities for percolation model in a random environment. Then, we
integrate out the randomness in the edge-parameters using a
multi-scale decomposition of the GFF. We believe that a similar
strategy could lead to proofs of the existence of a phase transition
for various other models.
Joint work with Hugo Duminil-Copin, Subhajit Goswami, Franco Severo,
Ariel Yadin.