The Geometry of Spherical Random Fields

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.