Claudio Durastanti, researcher at the Department of Basic and Applied Sciences for Engineering (SBAI) of Sapienza University, was our guest at the D2 Seminar Series presenting his talk on “Spherical Poisson Waves“.

During his talk, he discussed a model of Poisson random waves defined in the sphere, to study Quantitative Central Limit Theorems when both the rate of the Poisson process (that is, the expected number of the observations sampled at a fixed time) and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures.

What are we talking about when we talk about Spherical Poisson Waves?

A spherical Poisson random wave (or random eigenfunction) is the superposition of a random number of deterministic waves, centered at points uniformly distributed on the sphere.

What is the main question you aim to answer with your research?
Is it possible to establish quantitative central limit theorems with respect to both the random  amount of waves and their energy levels? A quantitative CLT is a CLT where the speed of convergence to Gaussianity is explicitly known.
What is the major challenge in addressing such a question?
The methods used to achieve our results involve the computation of several probabilistic bounds. Some of those calculations are complicate and involve a good knowledge of the harmonic analysis on the sphere and related properties.
Are there any practical applications of your work? Would you like to predict for us what may be the future outcomes of your research?
Our model is very close to the random phase model very popular in the physicists’ community where, roughly speaking, Gaussian random waves are commonly hinted as a universal model to approximate the behavior of deterministic eigenfunctions in the high-energy limit.
Our model can be read as the starting point of a new research line. The next step will consist in focusing on spherical Poisson waves defined over functional spaces which guarantee also finite distribution convergence, that is, the convergence of the coefficients characterized by the Fourier expansion of the wave.