Berry-Heisenberg Random Waves

Abstract

We construct a new family of random fields on the Heisenberg group $\mathbb{H}$, the sub-Riemannian analog of ${\mathbb{R}}^n$. These fields are generalized random eigenfunctions of the sub-Laplacian on $\mathbb{H}$, and can be viewed as the sub-Riemannian counterpart to the Berry random wave model in ${\mathbb{R}}^n$. The construction of such waves relies on the representation theory of $\mathbb{H}$, and differs from the Euclidean case because of the presence of infinite-dimensional unitary irreducible representations. This work represents a first step towards studying random waves and their geometry in sub-Riemannian spaces.