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Pré-Publication, Document De Travail Année : 2017

Quasi-independence for nodal lines

Alejandro Rivera
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Résumé

\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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Dates et versions

hal-01634287 , version 1 (13-11-2017)
hal-01634287 , version 2 (18-12-2017)
hal-01634287 , version 3 (07-03-2019)

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Alejandro Rivera, Hugo Vanneuville. Quasi-independence for nodal lines. 2017. ⟨hal-01634287v1⟩
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