Seminari del Dipartimento
Fisica Matematica Phase transition of Bernoulli percolation via the Gaussian Free Field H.Duminil-Copin 12-02-2019 - 14:30 Largo San Leonardo Murialdo,1 - Pal.C - Aula 211
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 relies on 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. |