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.