On Nonlinear Functionals of Random Spherical Eigenfunctions

Marinucci, Domenico; Wigman, Igor

doi:10.1007/s00220-014-1939-7

Cited by 47 publications

(93 citation statements)

References 31 publications

(67 reference statements)

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“…More explicitly (see also Marinucci & Rossi, ; Marinucci & Wigman, , ; Rossi, ), we have the following analytic expressions for the leading term components of the LKCs (expected values and dominant stochastic term):…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%

“…Analogous results, although with different constants, can be established on subdomains of the sphere (see Todino, ). For u = 0, we obtain a quantity equivalent to the so‐called defect (see Marinucci & Wigman, ), that is,

D_{ℓ} = 2 {scriptL}_{2} false(A_{u = 0} (f_{ℓ}; {double-struckS}^{2}) false) - 4 π;

the expected value is immediately seen to be zero, while it can be shown that the variance is given by

Var false(D_{ℓ} false) = \frac{C}{ℓ^{2}} + o false(\frac{1}{ℓ^{2}} false),

ℓ \to \infty

, where the constant C can be computed as

C = 32 π {false\sum}_{k = 1}^{\infty} a_{k} C_{2 k + 1} and a_{k} = \frac{(2 k)!}{4^{k} {(k!)}^{2} false(2 k + 1 false)}

(see Equation (25); Marinucci & Wigman, ), and

C_{q} : = \int_{0}^{L} J_{0} {(ψ)}^{q} ψ d ψ, for q = 3 and q \geq 5,

with

J_{0} false(x false) = {false\sum}_{k = 0}^{\infty} {(- 1)}^{k} x^{2<...>}

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%

“…More explicitly, it was shown in Marinucci and Wigman () that, as

ℓ \to \infty

\int_{0}^{1} P_{ℓ}^{4} false(t false) d t \sim \frac{3}{2 π^{2}} \frac{\log ℓ}{ℓ^{2}};

to find a better approximation, we evaluate numerically the constant

\underset{ℓ \to \infty}{falselim} false[\int_{0}^{L} J_{0}^{4} (ψ) ψ d ψ - \frac{3}{2 π^{2}} \log ℓ false] \approx 0.297,

see the Appendix for some analytic results. Hence, we conclude that, up to smaller order terms,

\begin{matrix} Var false(h_{ℓ; 4} false) \sim 4! {(4 π)}^{2} \frac{1}{ℓ^{2}} false{\frac{3}{2 π^{2}} \log ℓ + 0.297 false} \\ \approx 4! 16 π^{2} \frac{1}{ℓ^{2}} \frac{3}{2 π^{2}} false{\log ℓ + 0.297 \frac{2 π^{2}}{3} false} \\ \approx 576 \frac{1}{ℓ^{2}} false{\log ℓ + 1.9542 false} . \end{matrix}

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%

“…to the so-called defect (see Marinucci & Wigman, 2014), that is, the expected value is immediately seen to be zero, while it can be shown that the variance is given by as → ∞, where the constant C can be computed as (see Equation (25); Marinucci & Wigman, 2014), and with being the J 0 Bessel function. In the appendix, we perform a numerical investigation on the value of the constant C; more precisely, to obtain a precision of 1.0 × 10 −4 , it is sufficient to sum the terms in (20) until q = 20, obtaining the value…”

Section: Excursion Area (K = 2)mentioning

confidence: 99%

“…Hence, when u = 0, the leading term in (25) disappears and the nodal length is asymptotic to the sample trispectrum, namely where h ,4 = ∫ 2 H 4 (f (x))dx, which is logarithmic and hence we derive (27). To be clear, as given in Marinucci et al (2017), as → ∞, and, setting L: = + 1 2 , we have that More explicitly, it was shown in Marinucci and Wigman (2014) that, as → ∞, to find a better approximation, we evaluate numerically the constant see the Appendix for some analytic results. Hence, we conclude that, up to smaller order terms, Then, the variance of the scaled sample trispectrum M is asymptotically given by Finally, let us recall that these results, as in Marinucci et al (2017), Rossi (2015), Todino (2018b) and Wigman (2010), refer to the boundary length, not to the first Lipschitz-Killing curvature; there is hence a difference of a factor 2 in the expected value, and a factor 4 in the variance.…”

Section: (Half) the Boundary Length (K = 1)mentioning

confidence: 99%

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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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Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%