Exponential rarefaction of real curves with many components

Gayet, Damien; Welschinger, Jean-Yves

doi:10.1007/s10240-011-0033-3

Cited by 38 publications

(53 citation statements)

References 23 publications

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“…In [30] and [17], the subject changed since observables which cannot be computed locally were estimated, beginning with the number of components the zero set of the random function. In [20], higher Betti numbers were studied.…”

Section: Introductionmentioning

confidence: 99%

Percolation of random nodal lines

Beffara¹,

Gayet²

2017

Publ.math.IHES

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172

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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

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Section: Introductionmentioning

confidence: 99%

Percolation of random nodal lines

Beffara¹,

Gayet²

2017

Publ.math.IHES

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172

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“…The first, Corollary 1.10 of [Let18], proves that the probability that there are no components in a prescribed region decays polynomially fast. The second, Theorem 1 of [GW11], deals with the other extreme and proves that polynomials of degree d >> 1 whose number of nodal components is maximal up to a linear term in d are exponentially rare in d. We hope that the proof of Theorem 1.4 can be adapted in order to get the lower concentration part of (1.1) with n = √ d for Kostlan polynomials of degree d >> 1.…”

mentioning

confidence: 96%

Quasi-independence for nodal lines

Rivera¹,

Vanneuville

2019

Ann. Inst. H. Poincaré Probab. Statist.

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We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance (x, y) → κ(x − y), with only mild conditions on the regularity of κ. As a first application, we study percolation for nodal lines in the spirit of [BG16]. In the said article, Beffara and Gayet rely on Tassion's method ([Tas16]) to prove that, under some assumptions on κ, most notably that κ ≥ 0 and κ(x) = O(|x| −325 ), the nodal set satisfies a box-crossing property. The decay exponent was then lowered to 16 + ε by Beliaev and Muirhead in [BM18]. In the present work we lower this exponent to 4 + ε thanks to a new approach towards quasi-independence for crossing events. This approach does not rely on quantitative discretization. Our quasi-independence result also applies to events counting nodal components and we obtain a lower concentration result for the density of nodal components around the Nazarov and Sodin constant from [NS16].

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“…where G n,d denotes the normalization constant needed to satisfy (11). Next using Equation (9) for d(n, ) and simplifying,…”

Section: Examplesmentioning

confidence: 99%

On the number of connected components of random algebraic hypersurfaces

Fyodorov

Lerario

Lundberg

2015

Journal of Geometry and Physics

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We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}P^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. The classification due to E. Kostlan shows that specifying an invariant ensemble is equivalent to assigning a weight to each eigenspace of the spherical Laplacian. Fixing $n$, we consider a family of invariant ensembles (choice of eigenspace weights) depending on the degree $d$. Under a rescaling assumption on the eigenspace weights (as $d \rightarrow \infty$), we prove that the order of growth of $\mathbb{E} b_0(X)$ satisfies: $$\mathbb{E} b_{0}(X)=\Theta\left(\left[ \mathbb{E} b_0(X\cap \mathbb{R}P^1) \right]^{n} \right). $$ This relates the average number of components of $X$ to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial $f|_{\mathbb{R}P^1}.$ The proof requires an upper bound for $\mathbb{E} b_0(X)$, which we obtain by counting extrema using Random Matrix Theory methods from recent work of the first author, and it also requires a lower bound, which we obtain by a modification of the barrier method. We also provide a quantitative upper bound for the implied constant in the above asymptotic; for the real Fubini-Study model these estimates reveal super-exponential decay of the leading coefficient (in $d$) of $\mathbb{E} b_0(X)$ (as $n \rightarrow \infty$).Comment: 24 pages, 1 figure. Now published in the Journal of Geometry and Physic

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Exponential rarefaction of real curves with many components

Cited by 38 publications

References 23 publications

Percolation of random nodal lines

Percolation of random nodal lines

Quasi-independence for nodal lines

On the number of connected components of random algebraic hypersurfaces

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