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1 December 2020 Existence of phase transition for percolation using the Gaussian free field
Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi, Franco Severo, Ariel Yadin
Duke Math. J. 169(18): 3539-3563 (1 December 2020). DOI: 10.1215/00127094-2020-0036


In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension d>4 undergoes a nontrivial phase transition (in the sense that pc<1). 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 1, 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.


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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.


Received: 1 August 2019; Revised: 13 March 2020; Published: 1 December 2020
First available in Project Euclid: 1 December 2020

MathSciNet: MR4181032
Digital Object Identifier: 10.1215/00127094-2020-0036

Primary: 97K50
Secondary: 82B43

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 18 • 1 December 2020
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