Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Quasi-independence for nodal lines

Alejandro Rivera and Hugo Vanneuville

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Abstract

We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance $(x,y)\mapsto\kappa(x-y)$, with only mild conditions on the regularity of $\kappa$. As a first application, we study percolation for nodal lines in the spirit of (Publ. Math. Inst. Hautes Études Sci. 126 (2017) 131–176). In the said article, Beffara and Gayet rely on Tassion’s method (Ann. Probab. 44 (2016) 3385–3398) to prove that, under some assumptions on $\kappa$, most notably that $\kappa\geq0$ and $\kappa(x)=O(|x|^{-325})$, the nodal set satisfies a box-crossing property. The decay exponent was then lowered to $16+\varepsilon$ by Beliaev and Muirhead in (Comm. Math. Phys. 359 (2018) 869–913). In the present work we lower this exponent to $4+\varepsilon$ thanks to a new approach towards quasi-independence for crossing events. This approach does not rely on quantitative discretization. Our quasi-independence result also applies to events counting nodal components and we obtain a lower concentration result for the density of nodal components around the Nazarov and Sodin constant from (Zh. Mat. Fiz. Anal. Geom. 12 (2016) 205–278).

Résumé

On démontre un résultat de quasi-indépendance pour les lignes de niveau de champs gaussiens planaires stationnaires centrés de covariance $(x,y)\mapsto\kappa(x-y)$, sous de faibles conditions sur la régularité de $\kappa$. On applique d’abord ce résultat à l’étude de la percolation des lignes nodales dans l’esprit de (Publ. Math. Inst. Hautes Études Sci. 126 (2017) 131–176). Dans ledit article, Beffara et Gayet s’appuyent sur la méthode de Tassion (Ann. Probab. 44 (2016) 3385–3398) pour démontrer que sous certaines hypothèses sur $\kappa$, notamment que $\kappa\geq0$ et $\kappa(x)=O(|x|^{-325})$, l’ensemble nodal satisfait une propriété de croisement de boîtes. L’exposant de décroissance a plus tard été réduit à $16+\varepsilon$ par Beliaev et Muirhead dans (Comm. Math. Phys. 359 (2018) 869–913). Dans le présent article nous baissons cet exposant jusqu’à $4+\varepsilon$ grâce à une nouvelle approche pour la quasi-indépendance d’événements de croisement. Cette approche ne s’appuie pas sur une discrétisation quantitative. Notre résultat de quasi-indépendance s’applique aussi à des événements de comptage de composantes nodales et nous obtenons un résultat de concentration par en dessous de la densité de composantes nodales autour de la constante de Nazarov et Sodin de (Zh. Mat. Fiz. Anal. Geom. 12 (2016) 205–278).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1679-1711.

Dates
Received: 22 December 2017
Revised: 28 June 2018
Accepted: 4 September 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1569398882

Digital Object Identifier
doi:10.1214/18-AIHP931

Mathematical Reviews number (MathSciNet)
MR4010948

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Gaussian fields Percolation Quasi-independence Influences

Citation

Rivera, Alejandro; Vanneuville, Hugo. Quasi-independence for nodal lines. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1679--1711. doi:10.1214/18-AIHP931. https://projecteuclid.org/euclid.aihp/1569398882


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