We prove using a novel random matrix model that all right-angled Artin groups
have a sequence of finite dimensional unitary representations that strongly
converge to the regular representation. We deduce that this result applies also
to: the fundamental group of a closed hyperbolic manifold that is either three
dimensional or standard arithmetic type, any Coxeter group, and any
word-hyperbolic cubulated group. There are applications of this to the spectral
geometry of e.g. hyperbolic 3-manifolds.Comment: 35 pages, 1 figur