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Valentina Cammarota

Publications and source records attributed to Valentina Cammarota.

At least 19 recordsLinked to original sources
arXiv2023

Correlation Structure and Resonant Pairs for Arithmetic Random Waves

The geometry of Arithmetic Random Waves has been extensively investigated in the last fifteen years, starting from the seminal papers [RW08, ORW08]. In this paper we study the correlation structure among different functionals such as nodal length, boundary length of excursion sets, and the number of intersection of nodal sets with deterministic curves in different classes; the amount of correlation depends in a subtle fashion from the values of the thresholds considered and the symmetry properties of the deterministic curves. In particular, we prove the existence of resonant pairs of threshold values where the asymptotic correlation is full, that is, at such values one functional can be perfectly predicted from the other in the high energy limit. We focus mainly on the 2-dimensional case but we discuss some specific extensions to dimension 3.

math.PR↗
arXiv2023

A Quantitative Functional Central Limit Theorem for Shallow Neural Networks

We prove a Quantitative Functional Central Limit Theorem for one-hidden-layer neural networks with generic activation function. The rates of convergence that we establish depend heavily on the smoothness of the activation function, and they range from logarithmic in non-differentiable cases such as the Relu to $\sqrt{n}$ for very regular activations. Our main tools are functional versions of the Stein-Malliavin approach; in particular, we exploit heavily a quantitative functional central limit theorem which has been recently established by Bourguin and Campese (2020).

math.PR↗
arXiv2021

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 $I \subset \mathbb{R}$. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random $L^2$-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.

math.PR↗
arXiv2020

No repulsion between critical points for planar Gaussian random fields

We study the behaviour of the point process of critical points of isotropic stationary Gaussian fields. We compute the main term in the asymptotic expansion of the two-point correlation function near the diagonal. Our main result implies that for a 'generic' field the critical points neither repel nor attract each other. Our analysis also allows to study how the short-range behaviour of critical points depends on their index.

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↗
arXiv2019

Boundary effect on the nodal length for Arithmetic Random Waves, and spectral semi-correlations

We test M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions ("boundary-adapted arithmetic random waves"). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels. It is found that the precise nodal deficiency or surplus of the nodal length depends on arithmetic properties of the energy levels, in an explicit way. To obtain the said results we apply the Kac-Rice method for computing the expected nodal length of a Gaussian random field. Such an application uncovers major obstacles, e.g. the occurrence of "bad" subdomains, that, one hopes, contribute insignificantly to the nodal length. Fortunately, we were able to reduce this contribution to a number theoretic question of counting the "spectral semi-correlations", a concept joining the likes of "spectral correlations" and "spectral quasi-correlations" in having impact on the nodal length for arithmetic dynamical systems. This work rests on several breakthrough techniques of J. Bourgain, whose interest in the subject helped shaping it to high extent, and whose fundamental work on spectral correlations, joint with E. Bombieri, has had a crucial impact on the field.

math.PR↗
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↗
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

Nodal area distribution for arithmetic random waves

We obtain the limiting distribution of the nodal area of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3= \mathbb{R}^3/ \mathbb{Z}^3$ ($3$-dimensional 'arithmetic random waves'). We prove that, as the multiplicity of the eigenspace goes to infinity, the nodal area converges to a universal, non-Gaussian, distribution. Universality follows from the equidistribution of lattice points on the sphere. Our arguments rely on the Wiener chaos expansion of the nodal area: we show that, analogous to (Marinucci, Peccati, Rossi and Wigman, 2016), the fluctuations are dominated by the fourth-order chaotic component. The proof builds upon recent results in (Benatar and Maffiucci, 2017) that establish an upper bound for the number of non-degenerate correlations of lattice points on the sphere. We finally discuss higher-dimensional extensions of our result.

math.PR↗
arXiv2017

Two point function for critical points of a random plane wave

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.

math-ph↗
arXiv2016

A Quantitative Central Limit Theorem for the Euler-Poincaré Characteristic of Random Spherical Eigenfunctions

We establish here a Quantitative Central Limit Theorem (in Wasserstein distance) for the Euler-Poincaré Characteristic of excursion sets of random spherical eigenfunctions in dimension 2. Our proof is based upon a decomposition of the Euler-Poincaré Characteristic into different Wiener-chaos components: we prove that its asymptotic behaviour is dominated by a single term, corresponding to the chaotic component of order two. As a consequence, we show how the asymptotic dependence on the threshold level $u$ is fully degenerate, i.e., the Euler-Poincaré Characteristic converges to a single random variable times a deterministic function of the threshold. This deterministic function has a zero at the origin, where the variance is thus asymptotically of smaller order. Our results can be written as an asymptotic second-order Gaussian Kinematic Formula for the excursion sets of Gaussian spherical harmonics.

math.PR↗
arXiv2016

Multiple testing of local maxima for detection of peaks on the (celestial) sphere

We present a topological multiple testing scheme for detecting peaks on the sphere under isotropic Gaussian noise, where tests are performed at local maxima of the observed field filtered by the spherical needlet transform. Our setting is different from the standard Euclidean/large same asymptotic framework, yet highly relevant to realistic experimental circumstances for some important areas of application in astronomy. More precisely, we focus on cases where a single realization of a smooth isotropic Gaussian random field on the sphere is observed, and a number of well-localized signals are superimposed on such background field. The proposed algorithms, combined with the Benjamini-Hochberg procedure for thresholding p-values, provide asymptotic strong control of the False Discovery Rate (FDR) and power consistency as the signal strength and the frequency of the needlet transform get large. This novel multiple testing method is illustrated in a simulation of point-source detection in Cosmic Microwave Background radiation (CMB) data.

math.ST↗
arXiv2015

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.

math.PR↗
arXiv2015

Entrance and sojourn times for Markov chains. Application to $(L,R)$-random walks

In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.

math.PR↗
arXiv2014

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.

math-ph↗