Number variance for homogeneous determinantal processes on hyperbolic spaces
Pierre Lazag
TL;DR
The paper studies homogeneous determinantal point processes on δ-hyperbolic spaces with exponential ball growth, proving a nontrivial, volume-proportional lower bound on the variance of the number of points in a ball. By expressing the variance through a lunule-volume integral and leveraging the hyperbolic geometry of the underlying space, the authors show that the variance cannot be asymptotically negligible compared to the mean, i.e., the processes are not hyperuniform. The results unify continuous and discrete hyperbolic examples, including radial DPPs on Cayley trees and Bergman-projection-based hyperbolic models, highlighting how hyperbolicity enforces chaotic fluctuations despite strong correlation structure. The approach hinges on the lunule-based variance formula for homogeneous DPPs and explicit geometric bounds on ball intersections in hyperbolic spaces.
Abstract
We consider an abstract determinantal point process on a general non--elementary Gromov hyperbolic metric space governed by an orthogonal projection in the case when the space is homogeneous and the point process is invariant under isometries. We give a lower bound of the variance of the number of points inside a ball that is proportional to the volume of the ball. In particular, such point processes are never hyperuniform. Our result applies to the known examples of radial determinantal point processes on Cayley trees and on the standard hyperbolic spaces governed by Bergman projection kernels.