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

Cammarota, Valentina; Marinucci, Domenico

doi:10.1214/17-aop1245

Cited by 48 publications

(89 citation statements)

References 39 publications

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“…As was noted in Cammarota and Marinucci (), the Gaussian Kinematic Formula can be rewritten with a very similar expression to (14), that is:

\begin{matrix} Proj false[L_{k} (A_{u} false(f_{ℓ}; S^{2} false)) false| 0 false] = [\begin{matrix} 2 \\ k \end{matrix}] {(}{, \frac{λ_{ℓ}}{2})}^{(2 - k) / 2} H_{1 - k} false(u false) ϕ false(u false) \frac{1}{{false(2 π false)}^{false(2 - k false) false/ 2}} \int_{S^{2}} H_{0} false(f_{ℓ} (x) false) d x + b_{k} false(ℓ false), \end{matrix}

where

b_{k} false(ℓ false) = \{\begin{matrix} 2 (1 - Φ (u)) = O (1) & for k = 0, \\ 0 & for k = 1, 2 \end{matrix} .

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 98%

“…Here, and in the sequel, we use

Proj [. | q]

for the projection of random quantities on the so‐called Wiener chaoses of order q ; the latter are spaces generated by linear combinations of Hermite polynomials of order q , computed in

f_{ℓ}

and its derivatives (we refer to Cammarota and Marinucci (), Marinucci et al (), Nourdin and Peccati () and the references therein for more discussions and details). It is also important to notice that

\frac{λ_{ℓ}}{2} = P_{ℓ}^{'} false(1 false)

represents the derivative of the covariance function of random spherical harmonics at the origin, so that the term

\frac{λ_{ℓ}}{2} \int_{S^{2}} H_{2} false(f_{ℓ} (x) false) d x .

can be viewed as a (random) measure of the sphere induced by the Riemannian metric, somewhat in analogy with the interpretation given for the Gaussian Kinematic Formula on the expected value in the book by Adler and Taylor (); recall indeed that for eigenfunctions

f_{ℓ}

on the sphere

S^{2}

the term

{scriptL}_{2} false(S^{2} false)

which appears in (10) is exactly given by the area of the sphere with radius

{\{\frac{λ_{ℓ}}{2}\}}^{1 false/ 2}

, that is,

{scriptL}_{2} false(S^{2} false) = \frac{λ_{ℓ}}{2} \times 4 π = \frac{λ_{ℓ}}{2} \int_{S^{2}} H_{0} false(f_{ℓ} (x) false) d x .

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%

“…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%

“…Given these results, Cammarota and Marinucci () showed that the variances of LKCs are dominated by the variance of the second‐order Wiener chaos; indeed, for the Euler‐Poincaré characteristic the expected value and variance are given, for an interval

I \subset R

, by

E false[L_{0} (A_{I} false(f_{ℓ}; S^{2} false)) false] = \frac{2}{\sqrt{2 π}} ℓ false(ℓ + 1 false) \int_{I} false(t^{2} - 1 false) e^{- t^{2} / 2} d t

Var false[L_{0} (A_{I} false(f_{ℓ}; S^{2} false)) false] = \frac{ℓ^{3}}{8 π} {[\int_{I} false(- t^{4} + 4 t^{2} - 1 false) e^{- \frac{t^{2}}{2}} d t]}^{2} + O false(ℓ^{5 false/ 2} false),

and in particular for semi‐intervals of the form

I = [u, \infty)

one obtains

E false[L_{0} (A_{u} false(f_{ℓ}; S^{2} false)) false] = \sqrt{\frac{2}{π}} e^{- u^{2} / 2} u \frac{ℓ (ℓ + 1)}{2} + 2 false[1 - normalΦ (u) false]

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 99%

“…This is again a form of the so‐called “Berry's cancellation phenomenon”, which we have also discussed earlier for the Lipschitz‐Killing curvatures. Indeed, the behavior of critical points and saddles can be shown to be dominated by the second‐order chaotic component, which takes the form (see Cammarota & Marinucci, ).

\frac{λ_{ℓ}}{2} false[\int_{u}^{\infty} p_{3}^{a} (t) d t false] \frac{1}{2 π} \int_{S^{2}} H_{2} false(f_{ℓ} (x) false) d x

and similarly for saddles.…”

Section: Characterization Of Critical Points For Random Spherical Harmentioning

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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“…As was noted in Cammarota and Marinucci (), the Gaussian Kinematic Formula can be rewritten with a very similar expression to (14), that is:

\begin{matrix} Proj false[L_{k} (A_{u} false(f_{ℓ}; S^{2} false)) false| 0 false] = [\begin{matrix} 2 \\ k \end{matrix}] {(}{, \frac{λ_{ℓ}}{2})}^{(2 - k) / 2} H_{1 - k} false(u false) ϕ false(u false) \frac{1}{{false(2 π false)}^{false(2 - k false) false/ 2}} \int_{S^{2}} H_{0} false(f_{ℓ} (x) false) d x + b_{k} false(ℓ false), \end{matrix}

where

b_{k} false(ℓ false) = \{\begin{matrix} 2 (1 - Φ (u)) = O (1) & for k = 0, \\ 0 & for k = 1, 2 \end{matrix} .

…”

Section: Characterization Of Excursion Sets For Random Spherical Harmmentioning

confidence: 98%

“…Here, and in the sequel, we use

Proj [. | q]

for the projection of random quantities on the so‐called Wiener chaoses of order q ; the latter are spaces generated by linear combinations of Hermite polynomials of order q , computed in