Percolation phase transition is nontrivial for graphs with isoperimetric dimension higher than 3
Let $G$ be a bounded-degree infinite graph. The first step to study percolation on $G$ is to prove the nontriviality of the phase transition. Let $p_c$ be the critical parameter of bond percolation on $G$ to be the infimum value of $p$ that you have an infinite cluster almost surely. We prove that if the isoperimetric dimension of $G$ is higher than 3, 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 proving a differential inequality, we relate the existence of an infinite cluster in percolation in the random environment to that of percolation with a parameter $p<1$.
This talk is based on a joint work with H. Duminil-Copin, S. Goswami, F. Severo, and A. Yadin.