A Quantitative Central Limit Theorem for the Euler-Poincaré Characteristic of Random Spherical Eigenfunctions

Cammarota, Valentina; Marinucci, Domenico

doi:10.48550/arxiv.1603.09588

Cited by 7 publications

(24 citation statements)

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“…The dominance of the fourth-order chaos, and the consequent lower order of the variance, was observed also for the Euler Characteristic of excursion sets of random Laplace eigenfunctions on the 2-dimensional sphere [4]. The investigation of the general validity of such asymptotic behavior for other geometric functionals of arithmetic random waves is left for future research.…”

Section: 3mentioning

confidence: 76%

Nodal area distribution for arithmetic random waves

Cammarota

2019

Trans. Amer. Math. Soc.

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We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on T 3 = R 3 /Z 3 (3-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian, distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to [21], the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results in [1] that establish an upper bound for the number of non-degenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.

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Section: 3mentioning

confidence: 76%

Nodal area distribution for arithmetic random waves

Cammarota

2019

Trans. Amer. Math. Soc.

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“…That is, X(t) is a random spherical harmonic and a Laplacian eigenfunction with eigenvalue −ℓ(ℓ + 1) [14,15,56].…”

Section: 4mentioning

confidence: 99%

“…These random fields satisfy the Helmholtz partial differential equation and have a degenerate covariance between the field and its Hessian. On the sphere, these become random spherical harmonics, which have been widely studied [14,15,56] due to applications in physics and astronomy. We obtain results for this boundary case as well using a different technique involving the Helmholtz equation.…”

Section: Introductionmentioning

confidence: 99%

Expected number and height distribution of critical points of smooth isotropic Gaussian random fields

Cheng¹,

Schwartzman

2018

Bernoulli

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We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.

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“…A general interpretation of these results can be given quickly as follows (we refer to [24], [25], [11] for more discussions and details). The nodal length L ℓ of random eigenfunctions can be expanded, in the L 2 −sense, in terms of its q-th order chaotic components, to obtain the orthogonal expansion:…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

“…As a consequence, the variance of the nodal volume Z(T ℓ ) has smaller order O(log ℓ), in the high energy limit, with respect to the variance of boundary length at thresholds different from zero, which has been shown to be O(ℓ) (see for instance [31]). This phenomenon is known as "Berry's cancellation" [6]; it is known to occur on the torus [18] and on other geometric functionals of random eigenfunctions, see i.e., [12], [13], [11]. More precisely, as far as the torus is concerned, Rudnick and Wigman in [35] and Krishnapur, Kurlberg and Wigman in [18] studied the volume of the nodal line (denoted with L ℓ ) of random eigenfunctions ("arithmetic random waves") T 2 = R 2 /Z 2 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Nodal Lengths in Shrinking Domains for Random Eigenfunctions on $\mathbb{S}^2$

Todino

2018

Preprint

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We investigate the asymptotic behavior of the nodal lines for random spherical harmonics restricted to shrinking domains, in the 2-dimensional case: i.e., the length of the zero set, where B r ℓ is the spherical cap of radius r ℓ . We show that the variance of the nodal length is logarithmic in the high energy limit; moreover, it is asymptotically fully equivalent, in the L 2 -sense, to the "local sample trispectrum", namely, the integral on the ball of the fourth-order Hermite polynomial. This result extends and generalizes some recent findings for the full spherical case. As a consequence a Central Limit Theorem is established.

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A Quantitative Central Limit Theorem for the Euler-Poincaré Characteristic of Random Spherical Eigenfunctions

Cited by 7 publications

References 0 publications

Nodal area distribution for arithmetic random waves

Nodal area distribution for arithmetic random waves

Expected number and height distribution of critical points of smooth isotropic Gaussian random fields

Nodal Lengths in Shrinking Domains for Random Eigenfunctions on $\mathbb{S}^2$

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