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 $\mathrm{\Pi }=\{\mathrm{\parallel }z{\mathrm{\parallel }}^{\alpha }+g_z{\}}_{z\in {\mathbb{Z}}^d}$ where $(g_z{)}_{z\in {\mathbb{Z}}^d}$ are jointly Gaussian, $\alpha >0$, $d\in \mathbb{N}$, and $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is a norm. We develop a new technique to prove sufficient conditions for the deletion singularity of $\mathrm{\Pi }$, 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 $\mathrm{\Pi }$ that are independent of the dimension $d$ and the correlation of the $g_z$’s.