Existence of phase transition for percolation using the Gaussian Free Field

TL;DR

This paper proves the existence of a non-trivial phase transition in Bernoulli percolation on certain graphs with high isoperimetric dimension, using Gaussian Free Field techniques to relate connectivity probabilities to percolation parameters.

Contribution

It introduces a novel method linking Gaussian Free Field functionals to percolation connectivity, establishing phase transition existence on graphs with isoperimetric dimension greater than four.

Findings

Percolation on graphs with $d>4$ has a phase transition with $p_c<1$.

Critical point for percolation on certain infinite graphs is strictly less than 1.

New technique using GFF to analyze percolation models.

Abstract

In this paper, we prove that Bernoulli percolation on bounded degree 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. The proof relies on a new technique consisting in expressing certain functionals of the Gaussian Free Field (GFF) in terms of connectivity probabilities for 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.