Let $G$ be a bounded-degree infinite graph, and $p_c$ be the critical parameter of bond percolation on $G$. That is $p_c$ is the infimum of values of $p$ that you have an infinite cluster almost surely.
In this talk, we prove that if the isoperimetric dimension of $G$ is higher than 4, then $p_c(G)<1$. The theorem settles affirmatively two conjectures of Benjamini and Schramm. Notably, if $G$ is a transitive graph with super-linear growth, then $p_c(G) <1$. In particular, it implies that if $G$ is a Cayley graph of a finitely generated group without a finite index cyclic subgroup, then $p_c(G)<1$.
The proof of the theorem starts with the existence of an infinite cluster for percolation in a certain in-homogeneous random environment governed by the Gaussian free field. Then, by the help of a multiscale decomposition of GFF, we relate the existence of an infinite cluster in percolation in the random environment to that of percolation with a fix parameter $p<1$.
This talk is based on a joint work with H. Duminil-Copin, S. Goswami, F. Severo, and A. Yadin.