Fine-scale properties of random functions

Abstract

We study the monochromatic ensemble of random functions in the generality of a compact Riemannian manifold of any dimension. We prove equidistribution of local integrals at scales within a logarithmic factor of the optimal wave scale. On the two-dimensional sphere, we prove a limit theorem for the distribution of these integrals. We also study nodal domains, giving explicit (but embarrassing) lower bounds for the Nazarov-Sodin constant in dimension 2 and 3 and an estimate of the high-dimensional behaviour.