Percolation of random nodal lines

Beffara, Vincent; Gayet, Damien

doi:10.48550/arxiv.1605.08605

Cited by 3 publications

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“…In this article, we prove a quasi-independence result for level lines of planar Gaussian fields and present two applications of this result. First, we use it to revisit and generalize the results by Gayet and Beffara [BG16] who initiated the study of large scale connectivity properties for nodal lines and nodal domains of planar Gaussian fields. Second, we apply it to the study of the concentration of the number of nodal lines around the Nazarov and Sodin constant (the constant ν of Theorem 1 of [NS16]).…”

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confidence: 99%

“…Box crossing estimates for planar Gaussian fields. In [BG16], the authors give conditions under which such sets satisfy a box-crossing property at p = 0. We say that random sets satisfy a box-crossing property if for any quad (i.e.…”

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confidence: 99%

“…a topological rectangle with two opposite distinguished sides) Q there exists a positive constant c such that for any (potentially sufficiently large) scale s, there is a crossing of sQ between distinguished sides by the random set with probability larger than c. The study of the case p = 0 is natural since this is the level at which duality arises, see for instance Remark A.11 in our appendix. The most important conditions asked in [BG16] were some symmetry conditions, the fact that f is positively correlated (which means that the covariance function κ takes only non-negative values) and a sufficiently fast decay for κ(x) as |x| does to +∞, namely κ(x) = O |x| −325 . In [BM18], Beliaev and Muirhead have lowered the exponent 325 to any α > 16.…”

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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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Quasi-independence for nodal lines

Rivera¹,

Vanneuville

2019

Ann. Inst. H. Poincaré Probab. Statist.

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“…Remark 2.4 Theorem 2.3 of Beffara and Gayet was originally stated for centred Gaussian vectors in [6]. By using the trivial bound P(|X| < ε) ≤ ε for any Gaussian random variable X with arbitrary mean, where ever necessary, in the proof of Theorem 2.3 it trivially extends to any non-centred Gaussian vector.…”

Section: Level Sets Of Gaussian Fieldsmentioning

confidence: 97%

“…The super-level sets at level u of this Gaussian field, defined as P u := {x ∈ V : X(x) ≥ u}, is a spin model. To show clustering of P u , we shall use the following total-variation distance bound between Gaussian random vectors from [6].…”

Section: Level Sets Of Gaussian Fieldsmentioning

confidence: 99%

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

2018

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Central limit theorems for linear statistics of lattice random fields (including spin models) are usually proven under suitable mixing conditions or quasi-associativity. Many interesting examples of spin models do not satisfy mixing conditions, and on the other hand, it does not seem easy to show central limit theorem for local statistics via quasi-associativity. In this work, we prove general central limit theorems for local statistics and exponentially quasi-local statistics of spin models on discrete Cayley graphs with polynomial growth. Further, we supplement these results by proving similar central limit theorems for random fields on discrete Cayley graphs taking values in a countable space, but under the stronger assumptions of α-mixing (for local statistics) and exponential α-mixing (for exponentially quasi-local statistics). All our central limit theorems assume a suitable variance lower bound like many others in the literature. We illustrate our general central limit theorem with specific examples of lattice spin models and statistics arising in computational topology, statistical physics and random networks. Examples of clustering spin models include quasi-associated spin models with fast decaying covariances like the off-critical Ising model, level sets of Gaussian random fields with fast decaying covariances like the massive Gaussian free field and determinantal point processes with fast decaying kernels. Examples of local statistics include intrinsic volumes, face counts, component counts of random cubical complexes while exponentially quasi-local statistics include nearest neighbour distances in spin models and Betti numbers of sub-critical random cubical complexes.Keywords Clustering spin models · mixing random fields · central limit theorem · Cayley graphs · fast decaying covariance · Gaussian random field · determinantal point process · exponentially quasi-local statistics · cubical complexes · nearest-neighbour graphs Mathematics Subject Classification (2000) 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs · 60G60 Random fields · 60F05 Central limit theorems and other weak theorems · 60D05 Geometric probability and stochastic geometry.

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The critical threshold for Bargmann–Fock percolation

Rivera¹,

Vanneuville²

2020

Annales Henri Lebesgue

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In this article, we study the excursion sets D p = f −1 ([−p, +∞[) where f is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, f is the centered Gaussian field on R 2 with covariance (x, y) → exp(− 1 2 |x − y| 2 ). Alexander has proved that, if p ≤ 0, then a.s. D p has no unbounded component. We show that conversely, if p > 0, then a.s. D p has a unique unbounded component. As a result, the critical level of this percolation model is 0. We also prove exponential decay of crossing probabilities under the critical level. To show these results, we rely on a recent box-crossing estimate by Beffara and Gayet. We also develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.

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Percolation of random nodal lines

Cited by 3 publications

References 0 publications

Quasi-independence for nodal lines

Quasi-independence for nodal lines

Central Limit Theorem for Exponentially Quasi-local Statistics of Spin Models on Cayley Graphs

The critical threshold for Bargmann–Fock percolation

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