  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.
