Quasi-independence for nodal lines

Abstract : We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance $(x,y)\mapsto\kappa(x-y)$. As a first application, we study percolation for nodal lines in the spirit of [BG16]. In the said article, Beffara and Gayet rely on Tassion's method ([Tas16]) to prove that, under some assumptions on $\kappa$, most notably that $\kappa \geq 0$ 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 [BM17]. 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 [NS15].
Type de document :
Pré-publication, Document de travail
IF_PREPUB. 34 pages, 4 figures, minor changes introduced. 2017
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Contributeur : Alejandro Rivera <>
Soumis le : lundi 18 décembre 2017 - 18:03:15
Dernière modification le : vendredi 26 octobre 2018 - 10:27:43


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  • HAL Id : hal-01634287, version 2
  • ARXIV : 1711.05009


Alejandro Rivera, Hugo Vanneuville. Quasi-independence for nodal lines. IF_PREPUB. 34 pages, 4 figures, minor changes introduced. 2017. 〈hal-01634287v2〉



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