A new link between Bernoulli percolation and the Gaussian Free Field

Hugo Duminil-Copin, IHES
Fine Hall 214

In this talk, we prove that Bernoulli percolation on graphs with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in the sense that $p_c<1$). 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 from 1996. The proof relieson a new technique consisting in 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 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.