Existence of phase transition for percolation using the Gaussian free field
Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi, Franco Severo, Ariel Yadin
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 pc<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.