# Analysis, Geometry, and Dynamical Systems Seminar

### Existence of phase transition for percolation on general graphs

The first step in the study of percolation on a graph $G$ is proving that its critical point $p_c(G)$ for the emergence of an infinite cluster is nontrivial, that is, $p_c(G)\lt 1$. In this talk we prove that, if the isoperimetric dimension of a graph $G$ (with bounded degree) is strictly larger than $4$, then $p_c(G)\lt 1$. This settles a conjecture of Benjamini and Schramm saying that $p_c(G)\lt 1$ 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.