Local Universality for Zeros and Critical Points of Monochromatic Random Waves

Canzani, Yaiza; Hanin, Boris

doi:10.1007/s00220-020-03826-w

Cited by 22 publications

(42 citation statements)

References 17 publications

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“…One can see this asymptotics either as an infill statistics statement since the performed scaling is nothing but a zooming (see Canzani and Hanin, 2020), or as a high energy statement (see Nourdin et al, 2019) since the second spectral moment λ κ of ψ κ is such that λ κ = κ 2 λ and hence tends to +∞.…”

Section: Resultsmentioning

confidence: 98%

“…While studying the random billiards, Berry (2002) argued that in the microscopic scale several models as arithmetic random waves on the torus or spherical harmonics, although they verify some boundary conditions, converge towards an universal Gaussian model, which is called Berry's random waves model. Canzani and Hanin (2020) studied the universality phenomenon in general Riemannian manifolds. The reader can find results on arithmetic random waves defined on the flat torus (Cammarota, 2019;Dalmao et al, 2019) and on random spherical harmonics in Cammarota and Marinucci (2019); Fantaye et al (2019); Marinucci and Rossi (2021) and references therein, see also Rossi (2019) for a survey on both subjects.…”

Section: Introductionmentioning

confidence: 99%

“…In order to study the asymptotic variance and the limit distribution we restrict our attention to the isotropic case and we let the domain increase to the whole space. It can be shown that this is equivalent to consider a fixed domain and taking the high energy limit, see Canzani and Hanin (2020); Nourdin et al (2019) and Remark 3.6 below. We find different orders of magnitude for the variance and different limit distributions for different models.…”

Section: Introductionmentioning

confidence: 99%

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On 3-dimensional Berry’s model

Dalmao¹,

Estrade²,

León³

2021

ALEA

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This work aims to study the dislocation or nodal lines of 3D Berry's random waves model. Their expected length is computed both in the isotropic and anisotropic cases, being them compared. Afterwards, in the isotropic case the asymptotic variance and distribution of the length are studied as the domain grows to the whole space. We find different orders of magnitude for the variance and different limit distributions for different submodels. The study includes the Berry's monochromatic random waves, the Bargmann-Fock model and the Black-Body radiation.

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Section: Resultsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On 3-dimensional Berry’s model

Dalmao¹,

Estrade²,

León³

2021

ALEA

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“…Given a compact Riemannian manifold M , we will say that a random field X : M → C is a monochromatic random wave of frequency λ ∈ R if X satisfies, almost surely, the Helmholtz equation for the eigenvalue −λ 2 : ∆ M X = −λ 2 X, with ∆ M being the Laplace-Beltrami operator. 4 For reasons that we will explain later (see Section 2) in this paper we will take on S 3 the round metric of a sphere of radius 2, so that for each ℓ ∈ 1 2 N, the eigenfunctions are all those complex valued functions whose real and imaginary parts are the restriction of real homogeneous harmonic polynomials on R 4 , where the ones of degree 2ℓ are relative to the eigenvalue −ℓ(ℓ + 1), for all ℓ ∈ 1 2 N 5 . It is well known that any square integrable random field X : S 3 → C admits a spectral representation as a sum (1.1)…”

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confidence: 99%

Isotropic random spin weighted functions on $S^2$ vs isotropic random fields on $S^3$

Stecconi

2021

Preprint

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We show that an isotropic random field on SU (2) is not necessarily isotropic as a random field on S 3 , although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on S 3 is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree d is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range {− d 2 , . . . , d 2 }, each of which is isotropic in the sense of SU (2). Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified.In addition we will give an overview of the theory of spin weighted functions and Wigner D-matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators ðð and the horizontal Laplacian of the Hopf fibration S 3 → S 2 , in the sense of [4]. Contents MICHELE STECCONI The Gaussian case 31References 321 There is some ambiguity in the literature, regarding the sign of s. We will justify our choice in Section 2.4.2 With this symbol we denote diffeomorphisms.3 If s ∈ Z, then T ⊗s is a true tensor power of the tangent bundle T S 2 = T ⊗1 and SO(3) is isomorphic to the frame orthonormal bundle of S 2 .

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“…Such a discrepancy can be partially explained by comparing the underlying covariance functions of the models, which is nearly monotonically decaying in the Euclidean setting and periodically oscillating on the torus. In Canzani and Hanin (2020); Zelditch (2009), the authors study monochromatic random waves on a general smooth compact manifold, that is, Gaussian linear combinations of eigenfunctions associated with eigenvalues ranging in a short interval.…”

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confidence: 99%

Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

Notarnicola¹

2021

ALEA

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Benatar and Maffucci (2019) established an asymptotic law for the variance of the nodal surface of arithmetic random waves on the 3-torus in the high-energy limit. In a subsequent work, Cammarota (2019) proved a universal non-Gaussian limit theorem for the nodal surface. In this paper, we study the nodal intersection length and the number of nodal intersection points associated, respectively, with two and three independent arithmetic random waves of same frequency on the 3-torus. For these quantities, we compute their expected value, asymptotic variance as well as their limiting distribution. Our results are based on Wiener-Itô expansions for the volume and naturally complement the findings of Cammarota (2019). At the core of our analysis lies an abstract cancellation phenomenon applicable to the study of level sets of arbitrary Gaussian random fields, that we believe has independent interest.

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Local Universality for Zeros and Critical Points of Monochromatic Random Waves

Cited by 22 publications

References 17 publications

On 3-dimensional Berry’s model

On 3-dimensional Berry’s model

Isotropic random spin weighted functions on $S^2$ vs isotropic random fields on $S^3$

Fluctuations of nodal sets on the 3-torus and general cancellation phenomena

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