Random Spherical Harmonics Research Articles

On the correlation between critical points and critical values for random spherical harmonics

We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I ⊂ R I \subset \mathbb {R} . We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L 2 L^2 -norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics.

Random Lipschitz–Killing curvatures: Reduction Principles, Integration by Parts and Wiener chaos

In this survey we collect some recent results regarding the Lipschitz–Killing curvatures (LKCs) of the excursion sets of random eigenfunctions on the two-dimensional standard flat torus (arithmetic random waves) and on the two-dimensional unit sphere (random spherical harmonics). In particular, the aim of the present survey is to highlight the key role of integration by parts formulae in order to have an extremely neat expression for the random LKCs. Indeed, the main tool to study local geometric functionals of random waves on manifold is to exploit their Wiener chaos decomposition and show that (often), in the so-called high-energy limit, a single chaotic component dominates their behavior. Moreover, reduction principles show that the dominant Wiener chaotic component of LKCs of random waves’ excursion sets at threshold level u ≠ 0 u\ne 0 is proportional to the integral of H 2 ( f ) H_2(f) , f f being the random field of interest and H 2 H_2 the second Hermite polynomial. This will be shown via integration by parts formulae.

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 characterization 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.

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 relative 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 the work of Krishnapur et al. [Ann. Math. 177(2), 699–737 (2013)]} and Bombieri and Bourgain [Int. Math. Res. Not. (IMRN) 11, 3343–3407 (2015)] have opened a door for an active ongoing research on the nodal length of functions defined on surfaces of arithmetic flavor, such as the torus or the square. Furthermore, Bourgain’s work [J. Bourgain, Isr. J. Math. 201(2), 611–630 (2014)] on toral Laplace eigenfunctions, also appealing to spectral correlations, allowed for inferring deterministic results from their random Gaussian counterparts.

On the Correlation Between Nodal and Nonzero Level Sets for Random Spherical Harmonics

We study the correlation between the nodal length of random spherical harmonics and the length of a nonzero level set. We show that the correlation is asymptotically zero, while the partial correlation after removing the effect of the random L^2-norm of the eigenfunctions is asymptotically one.

Fluctuations in the number of nodal domains

We show that the variance of the number of connected components of the zero set of the two-dimensional Gaussian ensemble of random spherical harmonics of degree n grows as a positive power of n. The proof uses no special properties of spherical harmonics and works for any sufficiently regular ensemble of Gaussian random functions on the two-dimensional sphere with distribution invariant with respect to isometries of the sphere. Our argument connects the fluctuations in the number of nodal lines with those in a random loop ensemble on planar graphs of degree four, which can be viewed as a step toward justification of the Bogomolny–Schmit heuristics.

Moderate Deviation estimates for Nodal Lengths of Random Spherical Harmonics

We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020+) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.

The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics

Nous étudions le comportement asymptotique de la longueur nodale de fonctions propres aléatoires $f_{\ell}$ du Laplacien sphérique pour valeurs propres très élevés $\ell\rightarrow+\infty$, c’est-à-dire la longueur de leur ensemble de niveau zéro $f_{\ell}^{-1}(0)$. Nous démontrons que la longueur nodale est asymptotiquement équivalente, au sens de $L^{2}$, au « sample trispectrum », c’est-à-dire l’intégral de $H_{4}(f_{\ell}(x))$, le polynôme de Hermite d’ordre quatre évalué en $f_{\ell}$. Une conséquence de ce résultat est un Théorème Central Limite quantitatif (dans le sens de la distance de Wasserstein) pour la longueur nodale, quand l’énergie tend vers l’infini.

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⊂R for Gaussian spherical eigenfunctions fℓ, in the high energy regime where ℓ→∞. We show that these fluctuations are asymptotically equivalent to the centred L2-norm of fℓ 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.

A numerical investigation on the high‐frequency geometry of spherical random eigenfunctions

AbstractA 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.

Critical radius and supremum of random spherical harmonics

We first consider deterministic immersions of the $d$-dimensional sphere into high dimensional Euclidean spaces, where the immersion is via spherical harmonics of level $n$. The main result of the article is the, a priori unexpected, fact that there is a uniform lower bound to the critical radius of the immersions as $n\to\infty$. This fact has immediate implications for random spherical harmonics with fixed $L^{2}$-norm. In particular, it leads to an exact and explicit formulae for the tail probability of their (large deviation) suprema by the tube formula, and also relates this to the expected Euler characteristic of their upper level sets.

A quantitative central limit theorem for the excursion area of random spherical harmonics over subdomains of S2

In recent years, considerable interest has been drawn by the analysis of geometric functionals for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In this paper, we extend those results to proper subsets of the sphere S2, i.e., spherical caps, focussing, in particular, on the excursion area. Precisely, we show that the asymptotic behaviour of the excursion area is dominated by the so-called second-order chaos component and we exploit this result to establish a quantitative central limit theorem, in the high energy limit. These results generalize analogous findings for the full sphere; their proofs, however, require more sophisticated techniques, in particular, a careful analysis (of some independent interest) for smooth approximations of the indicator function for spherical cap subsets.

Quantum ergodicity of Wigner induced random spherical harmonics

We introduce a new notion of a ‘random orthonormal basis of spherical harmonics’ of L^(S^2) using generalized Wigner ensembles and show that such a random basis is almost surely quantum ergodic. On one hand, because of how our random basis is constructed, this generalizes previous results for random eigenfunctions that are defined using Haar measures on unitary groups. On the other hand, this provides a semi-classical realization of the probabilistic ‘local quantum unique ergodicity’ of Wigner eigenvectors.

Critical radius and supremum of random spherical harmonics (II)

We continue the study, begun in \cite{FA}, of the critical radius of embeddings, via deterministic spherical harmonics, of fixed dimensional spheres into higher dimensional ones, along with the associated problem of the distribution of the suprema of random spherical harmonics. Whereas \cite{FA} concentrated on spherical harmonics of a common degree, here we extend the results to mixed degrees, obtaining larger lower bounds on critical radii than we found previously.

Fluctuations of the total number of critical points of random spherical harmonics

We determine the asymptotic law for the fluctuations of the total number of critical points of random Gaussian spherical harmonics in the high degree limit. Our results have implications on the sophistication degree of an appropriate percolation process for modelling nodal domains of eigenfunctions on generic compact surfaces or billiards.

Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics

In this short note, we build upon recent results from our earlier paper to present a precise expression for the asymptotic variance of the Euler-Poincaré characteristic for the excursion sets of Gaussian eigenfunctions on S 2 \mathcal {S}^2 ; this result can be written as a second-order Gaussian kinematic formula for the excursion sets of random spherical harmonics. The covariance between the Euler-Poincaré characteristics for different level sets is shown to be fully degenerate; it is also proved that the variance for the zero level excursion sets is asymptotically of smaller order.

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.

On the Distribution of the Critical Values of Random Spherical Harmonics

We study the limiting distribution of critical points and extrema of random spherical harmonics, in the high-energy limit. In particular, we first derive the density functions of extrema and saddles; we then provide analytic expressions for the variances and we show that the empirical measures in the high-energy limits converge weakly to their expected values. Our arguments require a careful investigation of the validity of the Kac–Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed.

Erratum to: Fluctuations of the Nodal Length of Random Spherical Harmonics

and not as given. Though the main essence stays unimpaired, the leading constant in the main quantities in this paper were effected as a result of the mistake made; in particular the leading constant for the nodal length variance is consistent with Berry’s prediction (taking into account the symmetry of the spherical nodal lines), unlike what is claimed in the original version of the manuscript. That is, the original claim of Theorem 1.1,

The defect variance of random spherical harmonics

The defect of a function is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is evaluating the defect variance, asymptotically in the high-frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern cosmological data analysis.