(Non-)Hyperuniformity of Second Order Statistics of Point Processes

We investigate statistical properties of certain stationary point processes, namely determinantal processes with projection kernels and Gibbs point processes with superstable pair interactions. These are examples of hyperuniform and non-hyperuniform stationary point processes, respectively. We are interested in the variance of their second order statistics within a ball around the origin, and we study the asymptotic growth of this variance as the radius of the ball goes to infinity. It is shown that, generically, for both types of processes the variance is asymptotically proportional to the volume of the ball. In other words: the second order statistics of these point processes behave non-hyperuniform. For Gibbs processes with superstable interactions these results have an interesting application to the so-called inverse Henderson problem of statistical mechanics.
We also show that the structure factor (respectively the Bartlett spectral measure) of these Gibbs processes is strictly positive, while it is positive except for a simple zero at the origin for the determinantal processes.