The Geometry of Spherical Random Fields

TL;DR

This thesis explores the geometry of random fields on spheres and other manifolds, characterizing isotropic Gaussian fields, analyzing high-energy eigenfunctions, and introducing new methods for spin fields.

Contribution

It provides new characterizations of isotropic Gaussian fields, proves non-existence results for Brownian fields on certain groups, and develops a novel approach for spin random fields on the sphere.

Findings

Non-existence of Lévy's Brownian field on SO(3) and related groups.

Quantitative CLT results for geometric functionals of eigenfunctions.

Non-universal distribution of nodal length in arithmetic random waves.

Abstract

In this PhD Thesis we investigate the geometry of random fields on compact Riemannian manifolds, in particular the two-dimensional sphere. In the first part, we characterize isotropic Gaussian fields on homogeneous spaces of a compact group and then we prove the non-existence of P. L\'evy's Brownian field on the group SO(3), which moreover allows to extend the same kind of result to SO(n), SU(n) for n bigger than 3. In the second part, we investigate the high-energy behavior of random eigenfunctions on the (hyper)sphere and on the torus proving quantitative CLT results for some geometric functionals of those eiegenfunctions. A nice non-Central and non-Universal result is shown for nodal length distribution in the case of arithmetic random waves. Finally, we extend representation formulas obtained in the first part to the case of spin random fields on the sphere, introducing a new approach i.e., the pullback random field, that easily allows to study random sections of homogeneous vector bundles.