Transports of Stationary Random Measures: Asymptotic Variance, Hyperuniformity, and Examples

Kurzfassung

In this talk, we explore transports of stationary random measures on R^d. Under a suitable mixing assumption on two different transports of a single random measure we prove that the resulting random measures have the same asymptotic variance. An important consequence is a mixing criterion that ensures the persistence of the asymptotic variance under a transport. We pay special attention to the case of a vanishing asymptotic variance, known as hyperuniformity, which implies a suppression of long-range density fluctuations. Our approach enables us to rigorously establish hyperuniformity for many point processes and random measures that are relevant, among others, to random self-organization and material design. In particular, we construct a perturbation that turns any ergodic point process of finite intensity into a hyperuniform one.