Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes

The bulk scaling limit of eigenvalue distribution on the complex plane C of the complex Ginibre random matrices provides a determinantal point process (DPP). This point process is a typical example of disordered hyperuniform system characterized by an anomalous suppression of large-scale density fluctuations. As extensions of the Ginibre DPP, we consider a family of DPPs defined on the D-dimensional complex spaces C, D ∈ N, in which the Ginibre DPP is realized when D = 1. This one-parameter family (D ∈ N) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szegő kernels for the reduced Heisenberg group. For each D, using the modified Bessel functions, an exact and useful expression is shown for the local number variance of points included in a ball with radius R in R^2D ≃ C^D. We prove that any DPP in the Heisenberg family is in the hyperuniform state of class I, in the sense that the number variance behaves as R^2D−1 as R → ∞. Our exact results provide asymptotic expansions of the number variances in large R.