On the topology of Gaussian random zero sets

TL;DR

This paper investigates the asymptotic behavior of topological features of zero sets of Gaussian random fields, including knots, using geometric and topological methods, with implications for various physical and mathematical models.

Contribution

It provides new asymptotic laws for the topology of Gaussian zero sets, extending to knots and complex models, linking random fields with differential topology.

Findings

Asymptotic laws for the number and Betti numbers of zero sets

Application to random knots and complex models

Integration of geometric and topological techniques

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.