  This paper is dedicated to the study of the topologies and nesting
configurations of the components of the zero set of monochromatic random waves.
We prove that the probability of observing any diffeomorphism type, and any
nesting arrangement, among the zero set components is strictly positive for
waves of large enough frequencies. Our results are a consequence of building
Laplace eigenfunctions in Euclidean space whose zero sets have a component with
prescribed topological type, or an arrangement of components with prescribed
nesting configuration.
