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

TL;DR

This paper studies a family of determinantal point processes called the Heisenberg family, showing they are hyperuniform with specific variance behavior, extending known results from the Ginibre process to higher dimensions.

Contribution

It introduces the Heisenberg family of DPPs in higher dimensions and derives exact formulas and asymptotics for local number variances, demonstrating hyperuniformity in these systems.

Findings

Heisenberg family DPPs are hyperuniform of Class I.

Exact formulas for local number variance using Bessel functions.

Number variance scales as R^{2D-1} for large R.

Abstract

The bulk scaling limit of eigenvalue distribution on the complex plane ${\mathbb{C}}$ 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 ${\mathbb{C}}$, $D \in {\mathbb{N}}$, in which the Ginibre DPP is realized when $D=1$. This one-parameter family ($D \in {\mathbb{N}}$) of DPPs is called the Heisenberg family, since the correlation kernels are identified with the Szeg\H{o} 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 ${\mathbb{R}}^{2D} \simeq {\mathbb{C}}^D$. 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 $R \to \infty$. Our exact results provide asymptotic expansions of the number variances in large $R$.