arXiv subjects

Domenico Marinucci

Publications and source records attributed to Domenico Marinucci.

At least 37 records · Page 2Linked to original sources

arXiv2023

Spherical Poisson Waves

We introduce a model of Poisson random waves in $\mathbb{S}^{2}$ and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures.

math.PR↗

arXiv2023

Geometric Methods for Spherical Data, with Applications to Cosmology

This survey is devoted to recent developments in the statistical analysis of spherical data, with a view to applications in Cosmology. We will start from a brief discussion of Cosmological questions and motivations, arguing that most Cosmological observables are spherical random fields. Then, we will introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames. We will then focus on some specific issues, including tools and algorithms for map reconstruction (\textit{i.e.}, separating the different physical components which contribute to the observed field), geometric tools for testing the assumptions of Gaussianity and isotropy, and multiple testing methods to detect contamination in the field due to point sources. Although these tools are introduced in the Cosmological context, they can be applied to other situations dealing with spherical data. Finally, we will discuss more recent and challenging issues such as the analysis of polarization data, which can be viewed as realizations of random fields taking values in spin fiber bundles.

astro-ph.CO↗

arXiv2022

Fluctuations of Level Curves for Time-Dependent Spherical Random Fields

The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of $u$-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case $u = 0$, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a Central Limit Theorem can be established by Fourth-Moment Theorems and a Breuer-Major argument.

math.PR↗

arXiv2022

Geometry and topology of spin random fields

Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be viewed as random sections of the s-th complex tensor power of the tangent bundle of the 2-sphere. In this paper, we discuss how to characterize their expected geometry and topology. In particular, we investigate the asymptotic behaviour, under scaling assumptions, of general classes of geometric and topological functionals including Lipschitz-Killing Curvatures and Betti numbers for (properly defined) excursion sets; we cover both the cases of fixed and diverging spin parameters s. In the special case of monochromatic fields (i.e., spin random eigenfunctions) our results are particularly explicit; we show how their asymptotic behaviour is non-universal and we can obtain in particular complex versions of Berry's random waves and of Bargmann-Fock's models as subcases of a new generalized model, depending on the rate of divergence of the spin parameter s.

math.PR↗

arXiv2022

A novel Cosmic Filament catalogue from SDSS data

In this work we present a new catalogue of Cosmic Filaments obtained from the latest Sloan Digital Sky Survey (SDSS) public data. In order to detect filaments, we implement a version of the Subspace-Constrained Mean-Shift algorithm, boosted by Machine Learning techniques. This allows us to detect cosmic filaments as one-dimensional maxima in the galaxy density distribution. Our filament catalogue uses the cosmological sample of SDSS, including Data Release 16, so it inherits its sky footprint (aside from small border effects) and redshift coverage. In particular, this means that, taking advantage of the quasar sample, our filament reconstruction covers redshifts up to $z=2.2$, making it one of the deepest filament reconstructions to our knowledge. We follow a tomographic approach and slice the galaxy data in 269 shells at different redshift. The reconstruction algorithm is applied to 2D spherical maps. The catalogue provides the position and uncertainty of each detection for each redshift slice. We assess the quality of the detections with several metrics, which show improvement with respect to previous public catalogues obtained with similar methods. We also detect a highly significant correlation between our filament catalogue and galaxy cluster catalogues built from microwave observations of the Planck Satellite and the Atacama Cosmology Telescope.

astro-ph.CO↗

arXiv2021

Some Recent Developments on the Geometry of Random Spherical Eigenfunctions

A lot of efforts have been devoted in the last decade to the investigation of the high-frequency behaviour of geometric functionals for the excursion sets of random spherical harmonics, i.e., Gaussian eigenfunctions for the spherical Laplacian $Δ_{\mathbf{S}^2}$. In this survey we shall review some of these results, with particular reference to the asymptotic behaviour of variances, phase transitions in the nodal case (the \emph{Berry's Cancellation Phenomenon}), the distribution of the fluctuations around the expected values, and the asymptotic correlation among different functionals. We shall also discuss some connections with the Gaussian Kinematic Formula, with Wiener-Chaos expansions and with recent developments in the derivation of Quantitative Central Limit Theorems (the so-called Stein-Malliavin approach).

math.PR↗

arXiv2021

Flexible-bandwidth Needlets

We investigate here a generalized construction of spherical wavelets/needlets which admits extra-flexibility in the harmonic domain, i.e., it allows the corresponding support in multipole (frequency) space to vary in more general forms than in the standard constructions. We study the analytic properties of this system and we investigate its behaviour when applied to isotropic random fields: more precisely, we establish asymptotic localization and uncorrelation properties (in the high-frequency sense) under broader assumptions than typically considered in the literature.

math.PR↗

arXiv2020

Nodal deficiency of random spherical harmonics in presence of boundary

We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions along the equator. For this model we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relatively to the rotation invariant model of random spherical harmonics. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Laplace eigenfunctions, has made a crucial impact in the field and the current trends within. His works on the spectral correlations (Theorem 2.2 in Krishnapur, Kurlberg and Wigman (2013)) and joint with Bombieri (Bourgain and Bombieri (2015)) have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavour, like the torus or the square. Further, Bourgain's work on toral Laplace eigenfunctions (Bourgain (2014)), also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.

math-ph↗

arXiv2020

Lipschitz-Killing Curvatures for Arithmetic Random Waves

In this paper, we show that the Lipschitz-Killing Curvatures for the excursion sets of Arithmetic Random Waves (toral Gaussian eigenfunctions) are dominated, in the high-frequency regime, by a single chaotic component. The latter can be written as a simple explicit function of the threshold parameter times the centered norm of these random fields; as a consequence, these geometric functionals are fully correlated in the high-energy limit. The derived formulae show a clear analogy with related results on the round unit sphere and suggest the existence of a general formula for geometric functionals of random eigenfunctions on Riemannian manifolds.

math.PR↗

arXiv2020

On the Correlation of Critical Points and Angular Trispectrum for Random Spherical Harmonics

We prove a Central Limit Theorem for the Critical Points of Random Spherical Harmonics, in the High-Energy Limit. The result is a consequence of a deeper characterizations of the total number of critical points, which are shown to be asymptotically fully correlated with the sample trispectrum, i.e., the integral of the fourth Hermite polynomial evaluated on the eigenfunctions themselves. As a consequence, the total number of critical points and the nodal length are fully correlated for random spherical harmonics, in the high-energy limit.

math.PR↗

arXiv2020

Non-Universal Fluctuations of the Empirical Measure for Isotropic Stationary Fields on $\mathbb{S}^2 \times \mathbb{R}$

In this paper, we consider isotropic and stationary real Gaussian random fields defined on $\mathbb{S}^2\times\mathbb{R}$ and we investigate the asymptotic behavior, as $T\rightarrow +\infty$, of the empirical measure (excursion area) in $\mathbb{S}^2\times\mathbb{R}$ at any threshold, covering both cases when the field exhibits short and long memory, i.e. integrable and non-integrable temporal covariance. It turns out that the limiting distribution is not universal, depending both on the memory parameters and the threshold. In particular, in the long memory case a form of Berry's cancellation phenomenon occurs at zero-level, inducing phase transitions for both variance rates and limiting laws.

math.PR↗

arXiv2019

Asymptotics for Spherical Functional Autoregressions

In this paper, we investigate a class of spherical functional autoregressive processes, and we discuss the estimation of the corresponding autoregressive kernels. In particular, we first establish a consistency result (in sup and mean-square norm), then a quantitative central limit theorem (in Wasserstein distance), and finally a weak convergence result, under more restrictive regularity conditions. Our results are validated by a small numerical investigation.

math.ST↗

arXiv2019

A Numerical Investigation on the High-Frequency Geometry of Spherical Random Eigenfunctions

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, i.e ., 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 behaviour 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.

math-ph↗

arXiv2019

Point Source Detection and False Discovery Rate Control on CMB Maps

We discuss a new procedure to search for point sources in Cosmic Microwave background maps; in particular, we aim at controlling the so-called False Discovery Rate, which is defined as the expected value of false discoveries among pixels which are labelled as contaminated by point sources. We exploit a procedure called STEM, which is based on the following four steps: 1) needlet filtering of the observed CMB maps, to improve the signal to noise ratio; 2) selection of candidate peaks, i.e., the local maxima of filtered maps; 3) computation of \emph{p-}values for local maxima; 4) implementation of the multiple testing procedure, by means of the so-called Benjamini-Hochberg method. Our procedures are also implemented on the latest release of Planck CMB maps.

astro-ph.CO↗

arXiv2019

On the correlation between nodal and boundary lengths for random spherical harmonics

We study the correlation between the nodal length of random spherical harmonics and the measure of the boundary for excursion sets at any non-zero level. We show that the correlation is asymptotically zero, while the partial correlation after controlling for the random $L^2$-norm on the sphere of the eigenfunctions is asymptotically one.

math-ph↗

arXiv2018

Planck-scale distribution of nodal length of arithmetic random waves

We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral Quasi-Correlations, i.e. unusually small sums of lattice points lying on the same circle.

math-ph↗

arXiv2018

A Reduction Principle for the Critical Values of Random Spherical Harmonics

We study here the random fluctuations in the number of critical points with values in an interval $I\subset \mathbb{R}$ for Gaussian spherical eigenfunctions $\left\{f_{\ell }\right\} $, in the high energy regime where $\ell \rightarrow \infty $. We show that these fluctuations are asymptotically equivalent to the centred $L^{2}$-norm of $\left\{ f_{\ell }\right\} $ times the integral of a (simple and fully explicit) function over the interval under consideration. We discuss also the relationships between these results and the asymptotic behaviour of other geometric functionals on the excursion sets of random spherical harmonics.

math.PR↗

arXiv2017

The Asymptotic Equivalence of the Sample Trispectrum and the Nodal Length for Random Spherical Harmonics

We study the asymptotic behaviour of the nodal length of random $2d$-spherical harmonics $f_{\ell}$ of high degree $\ell \rightarrow\infty$, i.e. the length of their zero set $f_{\ell}^{-1}(0)$. It is found that the nodal lengths are asymptotically equivalent, in the $L^{2}$-sense, to the "sample trispectrum", i.e., the integral of $H_{4}(f_{\ell}(x))$, the fourth-order Hermite polynomial of the values of $f_{\ell}$. A particular by-product of this is a Quantitative Central Limit Theorem (in Wasserstein distance) for the nodal length, in the high energy limit.

math.PR↗