Percolation of random nodal lines

Abstract

International audienceWe 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

The sharp phase transition for level set percolation of smooth planar Gaussian fields

“…For the events Cross (2R, R), the required statement is a direct consequence of Theorem 5.1. Standard gluing arguments (see [BG17,Section 4.2] for details) allow the conclusion to be extended to every quad.…”

“…To do so, one would first discretise the field by restricting it to a lattice (as in [BG17] for instance), and view crossings of quads as paths on this lattice. One might then hope to apply the non-product OSSS inequality directly to the (finite) family of Gaussian variables f (v) that determine each crossing event.…”

“…Given the discussion above, we see that our results apply to this planar Gaussian field for each β > 2. On the other hand, the covariance kernel of this field satisfies κ(x) ∼ c|x| −β , as |x| → ∞, and so none of the previous results in the literature (for instance in [BG17,RV19b]) apply to this field unless β > 4, and even then only the RSW estimates of Theorem 1.11 were known [RV19b]. In particular, the results in Theorems 1.6 and 1.11 in the case > 0, and the result in Theorem 1.15, were not previously known for any value of β > 2.…”

The critical threshold for Bargmann–Fock percolation

“…If D 0 had a.s. an unbounded connected component, then, by symmetry (since f is centered) this would also be the case for R 2 \ D 0 . But this would imply that N 0 has an unbounded connected component, thus contradicting Theorem 2.2 of [Ale96].-More recently, Beffara and Gayet [BG16] have proved a more quantitative version of Theorem 1.1 which holds for a large family of positively correlated stationary Gaussian fields such that κ(x) = O(|x| −α ) for some α sufficiently large. In [RV17], the authors of the present paper have revisited the results by [BG16] and weaken the assumptions on α.…”

“…These sets have been studied through their connections to percolation theory (see [MS83a,MS83b,MS86], [Ale96], [BS07], [BG16], [BM18], [BMW17], [RV17]). In this theory, one wishes to determine whether of not there exist unbounded connected components of certain random sets.…”