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Percolation of random nodal lines

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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 \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) 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 \(\{x\in \smash{\bar{U}},\ f(sx) > 0\} \) intersecting both \(\gamma \) and \(\gamma '\), 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 \(\{x\in \smash{\bar{U}},\ f(sx) = 0\} \). 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.

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Authors and Affiliations

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

    Vincent Beffara & Damien Gayet

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  1. Vincent Beffara
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  2. Damien Gayet
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Correspondence to Damien Gayet.

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Beffara, V., Gayet, D. Percolation of random nodal lines. Publ.math.IHES 126, 131–176 (2017). https://doi.org/10.1007/s10240-017-0093-0

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