Variance linearity for real Gaussian zeros

Lachièze-Rey, Raphaël

doi:10.48550/arxiv.2006.10341

Cited by 6 publications

(7 citation statements)

References 17 publications

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“…This implies the positivity of σ 2 in Proposition 1.11. The present paper partially overlaps with [25] since we obtained independently a similar lower bound for σ 2 by the same method, see [25,Section 4] and Corollary 4.8 below.…”

Section: Related Workmentioning

confidence: 63%

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Zeros of smooth stationary Gaussian processes

Ancona,

Letendre

2020

Preprint

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Let f : R → R be a stationary centered Gaussian process. For any R > 0, let νR denote the counting measure of {x ∈ R | f (Rx) = 0}. In this paper, we study the large R asymptotic distribution of νR. Under suitable assumptions on the regularity of f and the decay of its correlation function at infinity, we derive the asymptotics as R → +∞ of the central moments of the linear statistics of νR. In particular, we derive an asymptotics of order R p 2 for the p-th central moment of the number of zeros of f in [0, R]. As an application, we derive a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures νR. More precisely, after a proper rescaling, νR converges almost surely towards the Lebesgue measure in weak- * sense. Moreover, the fluctuation of νR around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the k-point function of the zero point process of f , for any k 2. Our analysis yields two results of independent interest. First, we derive an equivalent of this k-point function near any point of the large diagonal in R k , thus quantifying the short-range repulsion between zeros of f . Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of f .

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Section: Related Workmentioning

confidence: 63%

“…In Section 4.2, we use the result of [23] to derive the lower bound on σ 2 mentioned in Remark 1.12. Let us mention that, very recently, Lachièze-Rey [25] proved that:…”

Section: Related Workmentioning

confidence: 99%

Zeros of smooth stationary Gaussian processes

Ancona,

Letendre

2020

Preprint

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“…The authors are not aware of any previously known rigid and ergodic process that is non-hyperuniform in dimensions d ≥ 2 (if W is the unit ball). An example for d = 1 has very recently been given in [12]. In this paper we will prove that the point process resulting from intersecting Poisson hyperplanes has very strong rigidity properties.…”

Section: Introductionmentioning

confidence: 92%

On strongly rigid hyperfluctuating random measures

Klatt¹,

Last²

2020

Preprint

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In contrast to previous belief, we provide examples of stationary ergodic random measures that are both hyperfluctuating and strongly rigid. Therefore, we study hyperplane intersection processes (HIPs) that are formed by the vertices of Poisson hyperplane tessellations. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in a bounded observation window grows faster than the size of the window. Here we show that the HIPs exhibit a particularly strong rigidity property. For any bounded Borel set B, an exponentially small (bounded) stopping set suffices to reconstruct the position of all points in B and, in fact, all hyperplanes intersecting B. Therefore, also the random measures supported by the hyperplane intersections of arbitrary (but fixed) dimension, are hyperfluctuating. Our examples aid the search for relations between correlations, density fluctuations, and rigidity properties.

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“…Elementary considerations yield that the average number of crossings on an interval is proportionnal to the length of the interval. Furthermore, if µ contains more than 1 (symmetrised) atom, the variance of the number of crossings is quadratic [17]. On R d , we rather focus on the Lebesgue measure of the nodal excursions…”

Section: Gaussian Excursionsmentioning

confidence: 99%

Diophantine Gaussian excursions and random walks

Lachièze-Rey

2021

Preprint

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We investigate the asymptotic variance of Gaussian nodal excursions in the Euclidean space, focusing on the case where the spectral measure has incommensurable atoms. This study requires to establish fine recurrence properties in 0 for the associated irrational random walk on the torus. We show in particular that the recurrence magnitude depends strongly on the diophantine properties of the atoms, and the same goes for the variance asymptotics of nodal excursions.More specifically, if the spectral measures contains atoms which ratios are well approximable by rationals, the variance is likely to have large fluctuations as the observation window grows, whereas the variance is bounded by the (d − 1)-dimensional measure of the window boundary if these ratio are badly approximable. We also show that, given any reasonable function, there are uncountably many sets of parameters for which the variance is asymptotically equivalent to this function.

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Variance linearity for real Gaussian zeros

Cited by 6 publications

References 17 publications

Zeros of smooth stationary Gaussian processes

Zeros of smooth stationary Gaussian processes

On strongly rigid hyperfluctuating random measures

Diophantine Gaussian excursions and random walks

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