Existence of phase transition for percolation using the Gaussian Free Field

Jeudi 22 novembre 2018 15:45-16:45 - Franco Severo - IHES

Résumé : The first step in the study of percolation on a graph $G$ is proving that its critical point for the emergence of an infinite connected component is nontrivial, that is $p_c(G)<1$. In this talk we prove that, if the isoperimetric dimension of a graph $G$ (with bounded degree) is strictly larger than 4, then $p_c(G)<1$. This settles a conjecture of Benjamini and Schramm saying that $p_c(G)<1$ for any transitive graph with super-linear growth.
The proof proceeds by first proving the existence of an infinite cluster for percolation with certain random edge-parameters induced by the Gaussian Free Field (GFF). Then we integrate out the randomness in the environment by using a multi-scale decomposition of the GFF.
Joint work with Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi and Ariel Yadin

Lieu : salle 3L15

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