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Article Dans Une Revue Publications Mathématiques de L'IHÉS Année : 2017

Percolation of random nodal lines

Vincent Beffara
Damien Gayet

Résumé

We prove a Russo-Seymour-Welsch 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 $\mathbb R^2$ 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 $\{x\in \bar U, \, f(sx) \textgreater{} 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 \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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Dates et versions

hal-01321096 , version 1 (24-05-2016)
hal-01321096 , version 2 (13-07-2016)
hal-01321096 , version 3 (25-09-2017)

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Citer

Vincent Beffara, Damien Gayet. Percolation of random nodal lines. Publications Mathématiques de L'IHÉS, 2017, 126 (1), pp.131-176. ⟨10.1007/s10240-017-0093-0⟩. ⟨hal-01321096v3⟩
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