This project focuses on the analysis of geometric feature of Gaussian random fields defined as random Laplace eigenfunctions on d-dimensional manifolds. On the sphere, in particular, the geometrical analysis of random Gaussian eigenfunctions emerge naturally form the analysis of the Fourier components of Gaussian fields and has recently drawn a big amount of interest in view of strong motivating applications in cosmology, image analysis and complexity of random energy landscapes.
The goal of this research project is to obtain developments in the study of the following geometrical features:
- the effect of nontrivial boundary conditions on the nodal structure both in the vicinity of the boundary and globally,
- the introduction of ad hoc comparison methods to measure the 'degree' of clustering of critical points and the short-range repulsion of critical points of different indexes,
- the development of a general variance formula for the Lipschitz-Killing curvatures for the excursion sets of random spherical eigenfunctions (i.e. Euler characteristic of the excursion sets, length of level curves, and excursion area) that could extend the well known Gaussian Kinematic Formula for the expected value of the Lipschitz-Killing curvatures,
- the statistical analysis of random fields and in particular the application of the theoretical results on the probabilistic behaviour of the geometric functional to introduce suitable statistics for testing non-Gaussianity and isotropy.
Berry [1] suggested that there exists a (non-rigorous) link between the (deterministic) Laplace eigenfunctions of a generic 'chaotic' manifold, and the random Laplace eigenfunctions on the plane restricted to a ball of radius of the order of the square root of eigenvalue. This vague relation, usually referred to as 'Berry's Random Wave Model' is subject to many numerical tests with overwhelmingly positive outcomes, see e.g. [2] and [3]. In [4], Berry argued that the presence of the boundary should impact the nodal length negatively and he derived the nodal deficiency for random Laplace eigenfunctions on the plane constrained to satisfy Dirichlet or Neumann boundary conditions. Our objective is investigating the effect of nontrivial boundary on the nodal structures of random Laplace eigenfunctions on other manifolds (starting with sphere and torus), either in the vicinity of the boundary, or globally. In this way we can test Berry's ansatz on nodal deficiency for Laplace eigenfunctions of generic 'chaotic' manifolds.
The study of critical points under the point of view of point processes and, in particular, their characterisation in terms of degree of clustering is a new approach that raises several interesting open problems. Intuitively a set of points is spatially homogeneous if approximately the same numbers of points occur in any spherical region of a given volume, and a set of points clusters if one observes points arranged in groups being well spaced out. The mathematical formalisation of such a statement appears not so easy and in the literature there are a few possible approaches: second-order statistics, void probabilities, moment measures, positive and negative association. Since the study of critical point is quite technically demanding only some of the approaches are computationally possible, this implies that it will be necessary to introduce both powerful computational tools (such as random matrices) and ad hoc comparison methods to extend the clustering comparison to the point process of critical points.
Up to our knowledge there are no results in the literature that go in the direction of deriving a Gaussian Kinematic Formula for the variance of the Euler characteristic of excursion sets of random Laplace eigenfunctions for generic manifolds. The derivation of such result would be a considerable achievement both from the point of view of theory and applications. From the point of view of the theory this could open the way to the understanding of higher moments of geometric functionals of Gaussian random fields. From the point of view of applications these results have potential for applications in the statistical analysis of random fields, for instance when testing for non-Gaussianity and isotropy or to search point-like sources/impurities in Cosmic Microwave Background radiation data as in the paper [5].
References
[1] M.V. Berry. Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12), 2083-2091 (1977)
[2] E. Bogomolny and C. Schmit. Percolation model for nodal domains of chaotic wave functions. Phys. Rev. Lett., no. 88 (2002)
[3] D. Belyaev and Z. Kereta. On the Bogomolny-Schmit conjecture. J. Phys. A 46, no. 45 (2013)
[4] M. V. Berry. Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A 35, 3025-3038 (2002)
[5] Y. Fantaye, V. Cammarota, D. Marinucci, A. P. Todino. A Numerical Investigation on the High-Frequency Geometry of Spherical Random Eigenfunctions. High Frequency, Volume 2, Issue 3-4 (2019)