Some properties of stationary determinantal point processes on $\mathbb{Z}$

TL;DR

This paper investigates stationary determinantal point processes on the integers, revealing their density, additive properties, and probabilistic inequalities, with extensions to higher dimensions.

Contribution

It establishes new properties of these processes, including Bohr-density, universality for convergence, and probabilistic inequalities, expanding understanding of their structure and behavior.

Findings

lmost surely Bohr-dense and universal for convergence.

emonstrates iscrepancy between syndeticity and additive coverage.

stablishes sub-Gaussian, Salem-Littlewood, and Khintchine-Kahane inequalities.

Abstract

We study properties of stationary determinantal point processes $\X$ on $\Z$ from different points of views. It is proved that $\X\cap \N$ is almost surely Bohr-dense and good universal for almost everywhere convergence in $L^1$, and that $\X$ is not syndetic but $\X +\X = \mathbb{Z}$. For the associated centered random field, we obtain a sub-Gaussian property, a Salem-Littlewood inequality and a Khintchine-Kahane inequality. Results can be generalized to $\Z^d$.