Abstract
In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension undergoes a nontrivial phase transition (in the sense that ). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasitransitive graphs (in particular, Cayley graphs) with superlinear growth is strictly smaller than , thus answering a conjecture of Benjamini and Schramm. The proof relies on a new technique based on expressing certain functionals of the Gaussian free field (GFF) in terms of connectivity probabilities for a percolation model in a random environment. Then we integrate out the randomness in the edge-parameters using a multiscale 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.
Citation
Hugo Duminil-Copin. Subhajit Goswami. Aran Raoufi. Franco Severo. Ariel Yadin. "Existence of phase transition for percolation using the Gaussian free field." Duke Math. J. 169 (18) 3539 - 3563, 1 December 2020. https://doi.org/10.1215/00127094-2020-0036
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