Existence of phase transition for percolation using the Gaussian free field

Abstract

In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a nontrivial phase transition (in the sense that $p_c<1$). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasitransitive graphs (in particular, Cayley graphs) with superlinear growth is strictly smaller than $1$, thus answering a conjecture of Benjamini and Schramm. The proof relies on a new technique based on expressing certain functionals of the Gaussian free field (GFF) in terms of connectivity probabilities for a percolation model in a random environment. Then we integrate out the randomness in the edge-parameters using a multiscale 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.