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29 March 2024 |
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Article overview
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Percolation of random nodal lines | Vincent Beffara
; Damien Gayet
; | Date: |
27 May 2016 | Abstract: | 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 ${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 | Services: | Forum | Review | PDF | Favorites |
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