Ricerc@Sapienza

The main novelty of this research project lies in the applications of the following probabilistic techniques to the derivations of the geometric properties of random fields:

i) the use of the so-called 'approximate Kac-Rice formula' [10] to perform variance computations for the geometric functionals under analysis,
ii) the idea of combining Kac-Rice formula and perturbation theory to derive explicit forms for the asymptotic variance of the geometric functionals in small domains,
iii) the idea of combining Kac-Rice formula and Morse theory to derive, from the study of critical points, results on the Euler characteristic of the excursions,
iv) the application of the powerful technique of Wiener-Itô chaos expansion and Stein-Malliavin methods to derive Central Limit Theorems for the geometric functionals of interest.

Critical points and nodal domains.
Nazarov and Sodin [8] proved that the expected number of nodal domains is asymptotic to the square root of the eigenvalue times a strictly positive constant 'a' in the high frequency limit. Little is known about the constant 'a'. Since the nodal domains number is bounded from above by the total number of critical points, this yields [38] an upper bound for 'a'. To the other hand no rigorous lower bound for the constant 'a' is known. Bogomolny and Schmit [7], and more recently Belyaev and Kereta [39], conjectured that in the high frequency limit, nodal domains are described by the clusters in a rectangular lattice bond percolation process and from this they derive some estimates on the constant 'a'. The variance of the total number of critical points derived the papers [27, 28] is crucial in determining the rigidity or flexibility of the percolation sites in the random percolation model proposed by Belyaev and Kereta (where the nodes are the local maxima and minima of the field). In this sense our goal is to derive the repulsion probability between two or more critical points. We plan to apply Kac-Rice formula to obtain an integral expression for factorial moments of the number of critical points in a domain and, via Perturbation Theory, we plan to investigate the asymptotic behavior of the Kac-Rice integral as the area of the domain becomes smaller and smaller.

A second order Gaussian Kinematic Formula.
In the paper [30] we present a full characterization for the asymptotic behavior for the variance of the first three Lipschitz-Killing curvatures for the excursion sets of random spherical eigenfunctions, i.e. the EPC, the length of level curves, and the excursion area. This seems to suggest the existence of a Gaussian Kinematic Formula for the variance. We plan to investigate the general validity of such variance's expressions for higher dimensional spheres and other compact Riemannian manifolds. We believe that the best strategy to attack the problem consists in performing a Wiener-Itô chaos expansion of the Lipschitz-Killing functionals and in studying the variance of the leading terms in such expansion.

Central Limit Theorems for geometric functionals.
Our results on the variance of the number of critical points [28] and Euler characteristic [30] open the way to the proof of a Central Limit Theorems for such random variables. We believe in fact that the number of critical points and the Euler characteristic of excursion sets obey a Gaussian limiting distribution in the high frequency limit. A key tool in the derivation of Central Limit Theorems is the analysis of the leading chaotic component in the chaos expansion, as well the fundamental result by Nourdin and Peccati [36] and some recent developments in Number Theory [40] on the angular distribution and correlations of lattice points on the sphere.

Gaussianity and isotropy tests.
One of the hypothesis at the base of the Big Bang is the assumption that Cosmic Microwave Background observations are realization of an isotropic, Gaussian random field. A natural question is whether the observed CMB maps are indeed consistent with these starting assumptions of Gaussianity and isotropy. Our results can be exploited in this setting by means of the implementation of a number of Gaussianity and isotropy tests. For instance, it is possible to compare the observed number of maxima in the data with its expected value and standard deviation under the hypothesis of Gaussianity and isotropy of the field, see e.g. [46].

Bibliography

[17] J. M. Azais and M. Wschebor. Level sets and extrema of random processes and fields. John Wiley & Sons Inc., Hoboken, NJ (2009)
[18] I. Nourdin and G. Peccati. Stein's Method on Wiener Chaos. Probability Theory and Related Fields, 145, no.1-2 (2009)
[19] D. Marinucci and G. Peccati. Random Fields on the Sphere. Representation, Limit Theorem and Cosmological Applications. Cambridge University Press (2011)
[20] N. Bartolo, E. Komatsu, S. Matarrese, A. Riotto. Non-Gaussianity from inflation: theory and observations. Phys. Rep. 402, no. 3-4 (2004)