On the Rigidity of Projected Perturbed Lattices

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.