The expected nodal volume of non-Gaussian random band-limited functions, and their doubling index

Abstract

The asymptotic law for the expected nodal volume of random non-Gaussian monochromatic band-limited functions is determined in vast generality. Our methods combine microlocal analytic techniques and modern probability theory. A particularly challenging obstacle that we need to overcome is the possible concentration of nodal volume on a small portion of the manifold, requiring solutions in both disciplines and, in particular, the study of the distribution of the doubling index of random band-limited functions. As for the fine aspects of the distribution of the nodal volume, such as its variance, it is expected that the non-Gaussian monochromatic functions behave qualitatively differently compared to their Gaussian counterpart. Some conjectures pertaining to these are put forward within this manuscript.