Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The sharp phase transition for level set percolation of smooth planar Gaussian fields

Stephen Muirhead and Hugo Vanneuville

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Abstract

We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two – roughly equivalent to the integrability of the covariance kernel – whereas previously the phase transition was only known in the case of the Bargmann–Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay exponentially.

Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion.

Résumé

Nous démontrons l’existence d’un phénomène de transition de phase pour les propriétés de connexion de lignes de niveau d’une grande classe de champs gaussiens planaires. En plus d’hypothèses de symétrie, régularié et positivité des corrélations, nous supposons que la covariance de ces champs décroît à vitesse polynomiale avec un exposant strictement plus grand que $2$. Nous montrons par ailleurs que la transition de phase est “nette” dans le sens que, dans la phase sous-critique, les probabilités de connexion convergent vers $0$ à vitesse exponentielle. Dans nos preuves, nous utilisons de façon centrale l’écriture des champs gaussiens lisses à l’aide d’un bruit blanc planaire. Nous utilisons cette représentation pour prouver de nouveaux résultats de quasi-indépendance spatiale, inspirés par la notion d’influence en théorie des fonctions booléennes. Par ailleurs, la structure produit du buit blanc nous permet d’utiliser l’inégalité d’OSSS, qui est une inégalité clef dans la récente approche d’étude des transitions de phase par Duminil-Copin, Raoufi et Tassion.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1358-1390.

Dates
Received: 25 January 2019
Accepted: 27 May 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345641

Digital Object Identifier
doi:10.1214/19-AIHP1006

Mathematical Reviews number (MathSciNet)
MR4076787

Subjects
Primary: 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Gaussian fields Percolation Sharp phase transition

Citation

Muirhead, Stephen; Vanneuville, Hugo. The sharp phase transition for level set percolation of smooth planar Gaussian fields. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1358--1390. doi:10.1214/19-AIHP1006. https://projecteuclid.org/euclid.aihp/1584345641


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