On the topology of Gaussian random zero sets

Abstract: We study the asymptotic laws for the number, Betti numbers, and isotopy classes of connected components of zero sets of real Gaussian random fields, where the random zero sets almost surely consist of submanifolds of codimension greater than or equal to one. Our results include `random knots' as a special case. Our work is closely related to a series of questions posed by Berry in [4,5]; in particular, our results apply to the ensembles of random knots that appear in the complex arithmetic random waves (Example 1.5), the Bargmann-Fock model (Example 1.1), Black-Body radiation (Example 1.2), and Berry's monochromatic random waves. Our proofs combine techniques introduced for level sets of random scalar-valued functions with methods from differential geometry and differential topology.