On the Rigidity of Projected Perturbed Lattices

TL;DR

This work develops a new approach to rigidity in projected perturbed lattices, analyzing point processes of the form $\Pi=\{V(z)+g_z\}_{z\in G}$ and establishing deletion singularity under a broad, dimension-free Assumption. The main theorem shows that finite deletions producing different cardinalities yield mutual singularity between corresponding processes, leading to deletion singularity when deleting all points outside the empty set. A key application to $\Pi=\{\|z\|^\alpha+g_z\}_{z\in\mathbb{Z}^d}$ demonstrates dimension-independent thresholds: deletion singularity for $\alpha>1$ in $\ell^1$ or $\ell^\infty$ norms, improving prior $\alpha> d/2$ bounds and extending to general $\ell^p$ norms. The results highlight a covariance-robust mechanism that can yield rigidity independent of spectral correlations, and they include optimality remarks and several illustrative examples to delineate the limits of the approach. Overall, the paper advances the understanding of rigidity phenomena in perturbed lattice models and provides tools potentially applicable to random Schrödinger eigenvalue problems.

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

Figures (1)

• Figure 1: For sake of clarity, we enumerate the elements of $G$ as $z_0,z_1,z_2,\ldots$, and we drop the dependence of $\Pi_T$ and $g_z$ on $\boldsymbol\omega$ in the figure above. The black dots on the top row represent the elements of $G\setminus S$, and the black dots on the bottom row represent the elements of $\Pi_T(\boldsymbol \omega)$; in both cases, red dots are deleted. (In the above example, we delete $B=\{z_1\}$ from $G$, and we delete $T=\{z_2,z_4\}$ from $\Pi$.) The arrows pointing downward represent the outputs of the bijection $\psi$ (e.g., $\psi(z_0)=V(z_0)+g_{z_0}(\boldsymbol \omega)$, and $\psi(z_2)=V(z_3)+g_{z_3}(\boldsymbol \omega)$). If we begin the algorithm described in the proof of Lemma \ref{['lem: f_k(Pi_T)>1 proof']} with the initial value $u_0=z_2\in T\setminus B$, then we get the finite sequence $(u_0,u_1,u_2)=(z_2,z_3,z_1)$. (This sequence can be visualized by following along the path traced by alternating dotted arrows and full arrows, starting from $V(u_0)+g_{u_0}(\boldsymbol \omega)$.) This sequence terminates at $n_\star=2$ because $u_2=z_1\in B$ is not mapped anywhere by $\psi$. Once that happens, we can now restart the algorithm with the new initial point $u_0'=z_4\in T\setminus B$. This point cannot have appeared in the previous sequence $u_0,u_1,u_2$, because it is not in the image of $\psi$. Moreover, Because all points in $B=\{z_1\}$ have been exhausted by the previous iteration of the algorithm, the algorithm is now guaranteed to generate an infinite sequence.

Theorems & Definitions (34)

• Definition 1.1
• Proposition 1.2: PeresSly
• Proposition 1.3
• Proposition 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10