Analysis and Geometry of Random Fields

Over the last decades, significant effort has been devoted to the investigation of the geometric and topological properties of random fields on manifolds, with particular emphasis on random eigenfunctions of the Laplace–Beltrami operator on Riemannian manifolds.

This book is based on the INdAM Workshop “Analysis and Geometry of Random Fields” held in Rome, Italy, on September 4-6, 2024. Over the last decades, significant effort has been devoted to the investigation of the geometric and topological properties of random fields on manifolds, with particular emphasis on random eigenfunctions of the Laplace–Beltrami operator on Riemannian manifolds. In the spherical setting, this probabilistic model was introduced by P. Bérard in 1985 to analyze the behavior of the nodal set of the "typical" eigenfunction, in the context of S.T. Yau’s 1982 conjecture. On the two-dimensional sphere, this model finds motivations in cosmology, specifically in connection with the cosmic microwave background, and also in mathematical physics, as it admits as a scale limit (when the eigenvalue tends to infinity) the well-known Berry random wave model. The latter is a random field on the Euclidean plane that, according to M. Berry’s 1977 conjecture, should predict the local behavior of (deterministic) eigenfunctions for billiards whose dynamics are classical and chaotic. There is a growing interest in extending results on fluctuations of geometric functionals from the two-dimensional sphere to more general manifolds, higher dimensions, and broader classes of spherical random fields, including those with temporal dependence. Such time-dependent random fields are of interest for applications in several disciplines, including climate sciences and Earth sciences. Finally, in the last few years, significant attention has been directed toward the connection between neural networks and random fields. This volume collects several contributions that advance the study of these topics.