Quasi-independence for nodal lines

Abstract : \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~\cite{bg_16}. In the said article, Beffara and Gayet rely on Tassion's method (\cite{tassion2014crossing}) 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 \cite{bm_17}. 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~\cite{nazarov2015asymptotic}.
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Pré-publication, Document de travail
IF_PREPUB. 2017
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Contributeur : Alejandro Rivera <>
Soumis le : lundi 13 novembre 2017 - 20:43:21
Dernière modification le : lundi 20 novembre 2017 - 12:48:24

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  • HAL Id : hal-01634287, version 1
  • ARXIV : 1711.05009

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Alejandro Rivera, Hugo Vanneuville. Quasi-independence for nodal lines. IF_PREPUB. 2017. 〈hal-01634287〉

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