Percolation of random nodal lines

Vincent Beffara; Damien Gayet

doi:10.1007/s10240-017-0093-0

Vincent Beffara ¹ ; Damien Gayet ¹

¹ Univ. Grenoble Alpes, CNRS, Institut Fourier 38000 Grenoble France

Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 131-176

Abstract

We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let $U$ be a smooth connected bounded open set in $R^{2}$ and $γ, γ^{'}$ two disjoint arcs of positive length in the boundary of $U$ . We prove that there exists a positive constant $c$ , such that for any positive scale $s$ , with probability at least $c$ there exists a connected component of the set ${x \in \bar{U}, f (s x) > 0}$ intersecting both $γ$ and $γ^{'}$ , where $f$ is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For $s$ large enough, the same conclusion holds for the zero set ${x \in \bar{U}, f (s x) = 0}$ . As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice.

Received: 2016-09-02
Accepted: 2017-09-09
Online First: 2017-09-18
Published online: 2017-11-07

MR Zbl

DOI: 10.1007/s10240-017-0093-0

Author's affiliations:

Vincent Beffara ¹; Damien Gayet ¹

¹ Univ. Grenoble Alpes, CNRS, Institut Fourier 38000 Grenoble France

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     author = {Vincent Beffara and Damien Gayet},
     title = {Percolation of random nodal lines},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {131--176},
     year = {2017},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {126},
     doi = {10.1007/s10240-017-0093-0},
     mrnumber = {3735866},
     zbl = {1412.60131},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-017-0093-0/}
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Vincent Beffara; Damien Gayet. Percolation of random nodal lines. Publications Mathématiques de l'IHÉS, Volume 126 (2017), pp. 131-176. doi: 10.1007/s10240-017-0093-0

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