Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics

Cammarota, Valentina; Marinucci, Domenico; Wigman, Igor

doi:10.1090/proc/13299

Cited by 30 publications

(43 citation statements)

References 21 publications

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“…The main purpose of this paper is to show that the high frequency behaviour is dominated (in the L 2 sense) by a single term with a very simple analytic expression, whose variance is indeed given by (6). In order to achieve this goal, we shall first establish the L 2 expansion of χ(A u (f ℓ ; S 2 )) into Wiener chaoses (see (16) below), which we will write as…”

Section: Resultsmentioning

confidence: 99%

“…More recently, a formula which can be viewed as an higher order extension of the Gaussian Kinematic Formula for the covariance of the Euler-Poincaré Characteristic characteristic of excursion sets at different thresholds, was established by [6], who focussed on an important class of fields: Gaussian spherical harmonics. Indeed, consider the Laplace equation…”

Section: Introductionmentioning

confidence: 99%

“…for a proof of formula (4) see, for example, [16], Corollary 5, or [10], Lemma 3.5. In [6], the results on the expected value were extended to an (asymptotic) evaluation of the variance; in particular, it was shown that, as ℓ → ∞…”

Section: Introductionmentioning

confidence: 99%

“…Asymptotic expressions for the variances of the two other Lipschitz-Killing curvatures for excursion sets in two dimensions, i.e. the area and (half) the boundary length, were also given in [17], [19], [18] and [23], [29]; in [6] all these expressions were collected in a unitary framework and it was conjectured that they could point out to a second-order extension of the Gaussian Kinematic Formula for random eigenfunctions. A further contribution in this direction is indeed given by our results in this paper, which we present below.…”

Section: Introductionmentioning

confidence: 99%

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A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

Cammarota¹,

Marinucci²

2018

Ann. Probab.

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We establish here a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincaré Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincaré Characteristic into different Wienerchaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level u is fully degenerate, i.e., the Euler-Poincaré Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics.•

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

Cammarota¹,

Marinucci²

2018

Ann. Probab.

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“…The Euler‐Poincaré characteristic (EPC) for random spherical harmonics was investigated by Cammarota and Marinucci () and Cammarota, Marinucci, and Wigman () among others, where the following expressions are given for the expected value and the second chaotic component:

\begin{matrix} Proj [{scriptL}_{0} false(A_{u} (f_{ℓ}; {double-struckS}^{2}) false) | 0] = \\ \{\frac{λ_{ℓ}}{2}\} [H_{1} false(u false) ϕ false(u false)] \frac{1}{2 π} \int_{S^{2}} H_{0} false(f_{ℓ} (x) false) d x + 2 \{1 - Φ (u)\}, \end{matrix}

\begin{matrix} Proj [{scriptL}_{0} false(A_{u} (f_{ℓ}; {double-struckS}^{2}) false) | 2] = \\ \frac{1}{2} \{\frac{λ_{ℓ}}{2}\} [H_{2} false(u false) H_{1} false(u false) ϕ false(u false)] \frac{1}{2 π} \int_{S^{2}} H_{2} false(f_{ℓ} (x) false) d x + O_{p} false(1 false) . \end{matrix}

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

et al. 2019

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A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high‐frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz‐Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler‐Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so‐called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.

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Bicovariograms and Euler characteristic of regular sets

Lachièze‐Rey

2017

Mathematische Nachrichten

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We establish an expression of the Euler characteristic of a -regular planar set in function of some variographic quantities. The usual  2 framework is relaxed to a  1,1 regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of ℝ 2 in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non-connected polyrectangular grains in ℝ 2 . Applications to excursions of smooth bivariate random fields are derived in the companion paper [25], and applied for instance to  1,1 Gaussian fields, generalising standard results. K E Y W O R D S Euler characteristic, intrinsic volumes, shot noise processes, boolean model M S C ( 2 0 1 0 ) 28A75, 52A22, 60D05, 60G10It is more generally an indicator of the regularity of the set, as an irregular structure is more likely to be shredded in many small pieces, or pierced by many holes, which results in a large value for | ( )|.As an integer-valued quantity, the Euler characteristic can be easily measured and used in estimation and modelisation procedures. It is an important indicator of the porosity of a random media [7,19,33], it is used in brain imagery [23,37], astronomy, [15,27,31], and many other disciplines. See also [1] for a general review of applied algebraic topology. In the study of parametric random media or graphs, a small value of | ( )| indicates the proximity of the percolation threshold, when that makes sense. See [30], or [13] in the discrete setting.The mathematical additivity property is expressed, for suitable sets and , by the formula ( ∪ ) = ( ) + ( ) − ( ∩ ), which applies recursively to finite such unions. In the validity domain of this formula, the Euler characteristic of a set can therefore be computed by summing local contributions. The Gauss-Bonnet theorem formalises this notion for  2 manifolds, stating that the Euler characteristic of a smooth set is the integral along the boundary of its Gaussian curvature. Exploiting the local nature of the curvature in applications seems to be a geometric challenge, in the sense that it is not always 398

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Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics

Abstract: In this short note, we build upon recent results from [7] to present a precise expression for the asymptotic variance of the Euler-Poincaré characteristic for the excursion sets of Gaussian eigenfunctions on S 2 .

Cited by 30 publications

References 21 publications

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

Bicovariograms and Euler characteristic of regular sets

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