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29 March 2024
 
  » arxiv » 1605.8605

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Percolation of random nodal lines
Vincent Beffara ; Damien Gayet ;
Date 27 May 2016
AbstractWe 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 ${xin ar U, , f(sx) extgreater{} 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 ${xin ar 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.
Source arXiv, 1605.8605
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