Existence of phase transition for percolation using the Gaussian Free Field

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.

19 pages. The current version has been accepted for publication in the Duke Mathematical Journal

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• Percolation for the Gaussian free field on the cable system: counterexamples
• Counting minimal cutsets and $p_c<1$
• A time-invariant random graph with splitting events
• Percolation phase transition on planar spin systems
• Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins