Random waves on sub-Riemannian $SU(2)$

Abstract

Random spherical eigenfunctions for the Lalplace-Beltrami operator are well understood and their local behavior, after renormalization, is described by Berry's random wave model. In this paper we construct a family of random fields on the sphere ${\mathbb{S}}^3$, identified with the group $SU(2)$, endowed with a sub-Riemannian structure. These fields are random eigenfunctions for the sub-Laplacian and they are constructed from matrix elements of unitary irreducible representations of $SU(2)$. We prove that the local behavior of such fields is the same as the Berry-Heisenberg random field on the Heisenberg group introduced in [10].