Existence of phase transition for percolation using the gaussian free field

Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi, Franco Severo, Ariel Yadin

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)3539-3563
Number of pages25
JournalDuke Mathematical Journal
Issue number18
StatePublished - 1 Dec 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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