Mathematics

An infinite point process (= an infinite discrete random set of points) is said to be hyperuniform along an exhaustion by bounded sets if the ratio (variance of the number of points)/(expectation of the number of points) goes to zero along the exhaustion. In this talk, I will present geometric methods that allow to derive hyperuniformity or non-hyperuniformity for determinantal point processes that are invariant by isometries. In particular, we obtain that translation invariant determinantal point processes on $R^d$ are hyperuniform along any exhaustion by open sets, while determinantal point processes on Gromov hyperbolic metric spaces - which include the known examples on Cayley trees and standard real or complex hyperbolic spaces $H^d$ - are never hyperuniform. If time permits, I will discuss applications to accumulated spectrograms together with some analogy with quantum ergodicity.