Spherical Random Field Research Articles

No smooth phase transition for the nodal length of band-limited spherical random fields

In this paper, we investigate the variance of the nodal length for band-limited spherical random waves. When the frequency window includes a number of eigenfunctions that grows linearly, the variance of the nodal length is linear with respect to the frequency, while it is logarithmic when a single eigenfunction is considered. Then, it is natural to conjecture that there exists a smooth transition with respect to the number of eigenfunctions in the frequency window; however, we show here that the asymptotic variance is logarithmic whenever this number grows sublinearly, so that the window “shrinks”. The result is achieved by exploiting the Christoffel–Darboux formula to establish the covariance function of the field and its first and second derivatives. This allows us to compute the two-point correlation function at high frequency and then to derive the asymptotic behaviour of the variance.

Geometric Methods for Cosmological Data on the Sphere.

This review is devoted to recent developments in the statistical analysis of spherical data, strongly motivated by applications in cosmology. We start from a brief discussion of cosmological questions and motivations, arguing that most cosmological observables are spherical random fields. Then, we introduce some mathematical background on spherical random fields, including spectral representations and the construction of needlet and wavelet frames. We then focus on some specific issues, including tools and algorithms for map reconstruction (i.e., separating the different physical components that 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 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. Expected final online publication date for the Annual Review of Statistics and Its Application, Volume 11 is March 2024. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Existence and Smoothness of Local Time and Self-intersection Local Time for Spherical Gaussian Random Fields

In this paper, we study properties of local time for spherical Gaussian random fields T on $${\mathbb{S}^2}$$ . For 2 < α < 4, we give a formally expression of local time for spherical Gaussian random fields, and obtain that the existence of local time in L2 if and only if (α − 2)d < 4 and the smoothness in the sense of Meyer–Watanabe if and only if (α − 2)(d + 2) < 4, respectively. Finally, the existence and smoothness of self-intersection local time of T are considered.

Fast and realistic large-scale structure from machine-learning-augmented random field simulations

ABSTRACT Producing thousands of simulations of the dark matter distribution in the Universe with increasing precision is a challenging but critical task to facilitate the exploitation of current and forthcoming cosmological surveys. Many inexpensive substitutes to full N-body simulations have been proposed, even though they often fail to reproduce the statistics of the smaller non-linear scales. Among these alternatives, a common approximation is represented by the lognormal distribution, which comes with its own limitations as well, while being extremely fast to compute even for high-resolution density fields. In this work, we train a generative deep learning model, mainly made of convolutional layers, to transform projected lognormal dark matter density fields to more realistic dark matter maps, as obtained from full N-body simulations. We detail the procedure that we follow to generate highly correlated pairs of lognormal and simulated maps, which we use as our training data, exploiting the information of the Fourier phases. We demonstrate the performance of our model comparing various statistical tests with different field resolutions, redshifts, and cosmological parameters, proving its robustness and explaining its current limitations. When evaluated on 100 test maps, the augmented lognormal random fields reproduce the power spectrum up to wavenumbers of $1 \, h \, \rm {Mpc}^{-1}$, and the bispectrum within 10 per cent, and always within the error bars, of the fiducial target simulations. Finally, we describe how we plan to integrate our proposed model with existing tools to yield more accurate spherical random fields for weak lensing analysis.

On multifractionality of spherical random fields with cosmological applications

This paper investigates spatial data on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying mathematical model of the cosmic microwave background (CMB) data. We discuss the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere. The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The results obtained suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, the CMB temperature intensities are multifractional. The methodology developed is used to suggest two approaches for detecting regions with anomalies in the cleaned CMB maps. doi:10.1017/S1446181122000104

Limiting behavior for the excursion area of band-limited spherical random fields

In this paper we investigate some geometric functionals for band-limited Gaussian and isotropic spherical random fields in dimension 2. In particular, we focus on the area of excursion sets, providing its behavior in the high energy limit. Our results are based on Wiener chaos expansion for non linear transform of Gaussian fields and on an explicit derivation on the high-frequency limit of the covariance function of the field. As a simple corollary we establish also the Central Limit Theorem for the excursion area.

Functional estimation of anisotropic covariance and autocovariance operators on the sphere

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of random fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of random fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover demonstrate the computational feasibility and practical merits of our estimation procedure in a simulation setting, assuming a fixed number of samples per random field. Our numerical estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory requirements by approximately three orders of magnitude compared to a naive implementation would require.

Spectral Analysis of Fractional Hyperbolic Diffusion Equations with Random Data

The paper studies the fundamental solutions to fractional in time hyperbolic diffusion equation or telegraph equations and their properties. Then it derives the exact solutions of the fractional hyperbolic diffusion equation with random data in terms of series expansions of isotropic in space spherical random fields on the unit sphere. Numerical illustration are presented to illustrate the theoretical results.

Increasing domain asymptotics for the first Minkowski functional of spherical random fields

The restriction to the sphere of a homogeneous and isotropic random field defines a spherical isotropic random field. This paper derives central and noncentral limit results for the first Minkowski functional subordinated to homogeneous and isotropic Gaussian and chi-squared random fields, restricted to the sphere in R3. Both scenarios are motivated by their interesting applications in the analysis of the Cosmic Microwave Background (CMB) radiation.

Topography of (exo)planets

Current technology is not able to map the topography of rocky exoplanets, simply because the objects are too faint and far away to resolve them. Nevertheless, indirect effect of topography should be soon observable thanks to photometry techniques, and the possibility of detecting specular reflections. In addition, topography may have a strong effect on Earth-like exoplanet climates because oceans and mountains affect the distribution of clouds \citep{Houze2012}. Also topography is critical for evaluating surface habitability \citep{Dohm2015}. We propose here a general statistical theory to describe and generate realistic synthetic topographies of rocky exoplanetary bodies. In the solar system, we have examined the best-known bodies: the Earth, Moon, Mars and Mercury. It turns out that despite their differences, they all can be described by multifractral statistics, although with different parameters. Assuming that this property is universal, we propose here a model to simulate 2D spherical random field that mimics a rocky planetary body in a stellar system. We also propose to apply this model to estimate the statistics of oceans and continents to help to better assess the habitability of distant worlds.

BaZr0.5Ti0.5O3: Lead-free relaxor ferroelectric or dipolar glass

Glassy freezing dynamics was investigated in ${\mathrm{BaZr}}_{0.5}{\mathrm{Ti}}_{0.5}{\mathrm{O}}_{3}$ (BZT50) ceramic samples by means of dielectric spectroscopy in the frequency range 0.001 Hz--1 MHz at temperatures $10lTl300$ K. From measurements of the quasistatic dielectric polarization in bias electric fields up to $\ensuremath{\sim}28$ kV/cm it has been found that a ferroelectric state cannot be induced, in contrast to the case of typical relaxors. This suggests that---at least for the above field amplitudes---BZT50 effectively behaves as a dipolar glass, which can be characterized by a negative value of the static third order nonlinear permittivity. The relaxation spectrum has been analyzed by means of the frequency-temperature plot, which shows that the longest relaxation time obeys the Vogel-Fulcher relation $\ensuremath{\tau}={\ensuremath{\tau}}_{0}\phantom{\rule{0.16em}{0ex}}\mathrm{exp}[{E}_{0}/(T\ensuremath{-}{T}_{0})]$ with the freezing temperature of 48.1 K, whereas the corresponding value for the shortest relaxation time is $\ensuremath{\sim}0$ K, implying an Arrhenius type behavior. By applying a standard expression for the static linear permittivity of dipolar glasses and/or relaxors the value of the Edwards-Anderson order parameter $q(T)$ has been evaluated. It is further shown that $q(T)$ can be described by the spherical random bond-random field model of relaxors.

High-frequency asymptotics for Lipschitz–Killing curvatures of excursion sets on the sphere

In this paper, we shall be concerned with geometric functionals and excursion probabilities for some nonlinear transforms evaluated on Fourier components of spherical random fields. In particular, we consider both random spherical harmonics and their smoothed averages, which can be viewed as random wavelet coefficients in the continuous case. For such fields, we consider smoothed polynomial transforms; we focus on the geometry of their excursion sets, and we study their asymptotic behaviour, in the high-frequency sense. We focus on the analysis of Euler-Poincar\'{e} characteristics, which can be exploited to derive extremely accurate estimates for excursion probabilities. The present analysis is motivated by the investigation of asymmetries and anisotropies in cosmological data. The statistics we focus on are also suitable to deal with spherical random fields which can only be partially observed, the canonical example being provided by the masking effect of the Milky Way on Cosmic Microwave Background (CMB) radiation data.

Cumulants on Wiener chaos: Moderate deviations and the fourth moment theorem

A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are considered.

A note on global suprema of band-limited spherical random functions

In this note, we investigate the behaviour of suprema for band-limited spherical random fields. We prove upper and lower bound for the expected values of these suprema, by means of metric entropy arguments and discrete approximations; we then exploit the Borell–TIS inequality to establish almost sure upper and lower bounds for their fluctuations. Band limited functions can be viewed as restrictions on the sphere of random polynomials with increasing degrees, and our results show that fluctuations scale as the square root of the logarithm of these degrees.

Dielectric properties of PMT-PT crystals

Results of broadband dielectric investigations of 0.94PbMg1/3Ta2/3O3–0.06PbTiO3 (0.94PMT-0.06PT or PMT-PT) crystals in wide temperature range from 100 K to 950 K are presented. Below 300 K the dielectric properties of crystals are governed by polar nanoregions dynamics. However, at higher temperatures (above 600 K) the electrical conductivity effects also become important. The electrical conductivity occurs presumably due the hopping of oxygen vacancies and demonstrates change in the activation energy close to 750 K. The change in the activation energy can be explained by increase of concentration of single ionized vacancies. No anomaly in the temperature dependence of the static dielectric permittivity was observed in wide temperature range from 213 K to 950 K and the dependence was successfully described by spherical random bonds random fields theory. Moreover, two different contributions were successfully separated in the distributions of relaxation times in PMT-PT crystals. These contributions were attributed to different polar nanoregions dynamics. Moreover, the different freezing temperatures values were obtained for most probable and longest relaxation times, which was explained with an idea of continuous distribution of relaxation times broadening on cooling in relaxors. The critical remarks about the Vogel-Fulcher law application for relaxors are addressed in the paper.

The stochastic properties of [formula omitted]-regularized spherical Gaussian fields

Convex regularization techniques are now widespread tools for solving inverse problems in a variety of different frameworks. In some cases, the functions to be reconstructed are naturally viewed as realizations from random processes; an important question is thus whether such regularization techniques preserve the properties of the underlying probability measures. We focus here on a case which has produced a very lively debate in the cosmological literature, namely Gaussian and isotropic spherical random fields, and we prove that neither Gaussianity nor isotropy are conserved in general under convex regularization based on ℓ1 minimization over a Fourier dictionary, such as the orthonormal system of spherical harmonics.

On Nonlinear Functionals of Random Spherical Eigenfunctions

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.

Gaussian semiparametric estimates on the unit sphere

We study the weak convergence (in the high-frequency limit) of the parameter estimators of power spectrum coefficients associated with Gaussian, spherical and isotropic random fields. In particular, we introduce a Whittle-type approximate maximum likelihood estimator and we investigate its asympotic weak consistency and Gaussianity, in both parametric and semiparametric cases.

Research on polarization effect for relaxor ferroelectrics by spherical random bond-random field model

Based on spherical random bond-random field model, micro-macro domain mechanism under electric field, and the fuzzy domain boundary with fractal dimension of electric field, the mechanism of polarization effect is analyzed. The results show that the effect of electric field on polarization effect of increment of domain dipoles leads to an unsaturated electric hysteresis loop and its associated big electrostrictive effect. But the variation of binding energy, when dipole couples to the increment of domain dipoles induced by electric field, has a little influence on electric hysteresis loop in a low electric field, and has almost no influence in a high electric field. The initial size of the micro domain is very important for electric hysteresis loop: small micro domain can lead to a long electric hysteresis loop and better linear relationship between electric field and electrostriction.

Weber and Beltrami integrals of squared spherical Bessel functions: finite series evaluation and high‐index asymptotics

Weber integrals and Beltrami integrals are studied, which arise in the multipole expansions of spherical random fields. These integrals define spectral averages of squared spherical Bessel functions with Gaussian or exponentially cut power‐law densities. Finite series representations of the integrals are derived for integer power‐law index μ, which admit high‐precision evaluation at low and moderate Bessel index n. At high n, numerically tractable uniform asymptotic approximations are obtained on the basis of the Debye expansion of modified spherical Bessel functions in the case of Weber integrals. The high‐n approximation of Beltrami integrals can be reduced to Legendre asymptotics. The Airy approximation of Weber and Beltrami integrals is derived as well, and numerical tests are performed over a wide range of Bessel indices by comparing the exact finite series expansions of the integrals with their high‐index asymptotics. Copyright © 2013 John Wiley &amp; Sons, Ltd.