On the Rigidity of Projected Perturbed Lattices

TL;DR

This paper investigates the rigidity and deletion singularity properties of projected perturbed lattices, introducing new techniques that improve understanding of these phenomena beyond traditional methods, with dimension-independent bounds.

Contribution

It develops a novel approach to establish sufficient conditions for deletion singularity in projected perturbed lattices, surpassing standard rigidity techniques and providing the first dimension-independent bounds.

Findings

Established new lower bounds on $oldsymbol{ ext{alpha}}$ for deletion singularity.

Introduced a technique that surpasses variance-based methods in rigidity analysis.

Provided the first dimension-independent conditions for deletion singularity.

Abstract

We study the occurrence of number rigidity and deletion singularity in a class of point processes that we call {\it projected perturbed lattices}. These are generalizations of processes of the form $\Pi=\{\|z\|^\alpha+g_z\}_{z\in\mathbb{Z}^d}$ where $(g_z)_{z\in\mathbb{Z}^d}$ are jointly Gaussian, $\alpha>0$, $d\in\mathbb{N}$, and $\|\cdot\|$ is a norm. We develop a new technique to prove sufficient conditions for the deletion singularity of $\Pi$, which improves significantly on the conditions one can obtain using the standard rigidity toolkit (e.g., the variance of linear statistics). In particular, we obtain the first lower bounds on $\alpha$ for the deletion singularity of $\Pi$ that are independent of the dimension $d$ and the correlation of the $g_z$'s.