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2021 Talagrand’s inequality in planar Gaussian field percolation
Alejandro Rivera
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Electron. J. Probab. 26: 1-25 (2021). DOI: 10.1214/21-EJP585


Let f be a stationary isotropic non-degenerate Gaussian field on R2. Assume that f=qW where qL2(R2)C2(R2) and W is the L2 white noise on R2. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that qq is pointwise non-negative and has fast enough decay, the set {f} percolates with probability one when >0 and with probability zero if 0. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcentration formula for the threshold T(g) which is the minimal value such that {gT(g)} contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.


The ideas of this paper stemmed from previous collaborations with Dmitry Beliaev, Stephen Muirhead and Hugo Vanneuville. I am grateful to the three of them for many helpful discussions. I am also thankful to Christophe Garban and Hugo Vanneuville for their comments on a preliminary version of this manuscript. Finally, I would like to thank the referee for their comments. I hope the revised version has improved in clarity.


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Alejandro Rivera. "Talagrand’s inequality in planar Gaussian field percolation." Electron. J. Probab. 26 1 - 25, 2021.


Received: 21 October 2019; Accepted: 30 January 2021; Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1214/21-EJP585

Primary: 60G60 , 60K35
Secondary: 82B43 , 82C43

Keywords: Gaussian fields , percolation , phase transition


Vol.26 • 2021
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