Probabilisitic limit theorems and the geometry of random fields

The topics presented in this thesis lie at the interface of probability theory and stochastic geometry, with emphasis on the asymptotic study of geometric objects associated with Gaussian random fields defined on manifolds. Such a research stream has been rapidly growing in past years, resulting in a number of developments focussing on local and global geometric quantities.
Our principal aim is to discuss probabilistic methods allowing one to deal with the asymptotic fluctuations of volumes of zero sets (also called nodal sets) of Gaussian Laplace eigenfunctions as the eigenvalues diverge to infinity, with a particular focus on the models of Arithmetic Random Waves on the three-dimensional torus and Berry's Random Plane Wave on the two-dimensional Euclidean space.
We prove universal variance estimates and non-Gaussian limit theorems for the zero sets of multiple independent Arithmetic Random Waves, complementing several related works in the literature.
Our analysis for this builds on an abstract cancellation result applicable to the setting of Gaussian Laplace eigenfunctions on manifolds, yielding in particular a formal description of the so-called Berry's Cancellation Phenomenon observed in various models of random eigenfunctions.
For the Berry Random Plane Wave, we prove spatial functional limit theorems for discretized and truncated versions of the nodal length indexed by rectangular domains. Such a contribution yields a basis for proving a fully general functional limit theorem for the nodal length and opens doors to a number of novel probabilistic limit theorems involving semi-local functionals of these nodal length processes.
A common technique lying at the core of our arguments for dealing with these tasks is the asymptotic analysis of the Wiener-Itô chaos expansion in Hermite polynomials of such geometric quantities, often allowing one to reduce investigations on Wiener chaoses of lower order. In this context, we discuss properties of generalized Hermite polynomials with matrix arguments, appearing in multivariate statistics and the theory of zonal polynomials. We argue that this family of orthogonal polynomials is particularly effective for deducing chaotic expansions of random variables that are symmetric functionals in the eigenvalues of underlying Gaussian random matrices, notably appearing in different applications dealing with the geometry of random fields.
We furthermore present a new characterization of matrix-Hermite polynomials as the eigenfunctions of a generalized Ornstein-Uhlenbeck semigroup on matrix spaces. The above mentioned probabilistic limit theorems originate from the systematic use of the Malliavin-Stein approach on Gaussian spaces, a collection of analytic statements allowing one to deduce probabilistic limit theorems by means of variational techniques. Such a series of results typically emerges from the combination of Stein's method for probabilistic approximations and Malliavin's infinite-dimensional differential calculus. At the end of this thesis, we present preliminary computations yielding variance estimates and Central Limit Theorems for certain non-linear functionals associated with the d-dimensional Berry Random field. Also, we discuss several aspects around optimal convergence rates within Gamma approximations of functionals of Gaussian fields.