Rigidity of random stationary measures and applications to point processes

TL;DR

This paper establishes a mathematical link between the zeros of the structure factor of stationary point processes and their rigidity properties, providing new criteria for understanding when such processes are determined by their outside configuration.

Contribution

It introduces a novel criterion connecting the zeros of the structure factor to k-rigidity in stationary point processes, extending previous results and applying to various models including determinantal processes.

Findings

k-rigidity occurs if the structure factor's continuous component has a zero of order k at 0

In continuous settings, local conditions are necessary for rigidity when the structure factor has finitely many zeros

Determinantal point processes exhibit k-rigidity if a specific spectral condition on the kernel's Fourier transform is met

Abstract

The {\it number rigidity} of a stationary point process $\mathsf{P}$ entails that for a bounded set $A$ the knowledge of $\mathsf{P}$ on $A^{c}$ a.s. determines $\mathsf{P}(A)$; the $k$-order rigidity means the moments of $\mathsf{P}1_{A}$ up to order $k$ can be recovered. We show that $k$-rigidity occurs if the continuous component $\mathscr{s}$ of $\mathsf{P}$'s {\it structure factor} has a zero of order $k$ in $0$, by exploiting a connection with Schwartz's Paley-Wiener theorem for analytic functions of exponential type; these results apply to any random $L^{2}$ wide sense stationary measure on $\mathbb{R}^{d}$ or $\mathbb{Z}^{d}$. In the continuous setting, these local conditions are also necessary if $\mathscr{s}$ has finitely many zeros, or is isotropic, or at the opposite separable. This explains why no model seems to exhibit rigidity in dimension $d\geqslant 3$, and allows to efficiently recover many recent rigidity results about point processes. For a field on $\mathbb{Z} ^{d}$, these results hold provided $\# A >2k$. For a continuous Determinantal point process with reduced kernel $\kappa$, $k$-rigidity is equivalent to $(1- \widehat {\kappa ^{2}})^{-1}$ having a zero of order $k$ in $0$, which answers questions on completeness and number rigidity. We also deduce some non-integrability results in the less tractable realm of Riesz gases. Finally, we are able to prove that random stationary quasicrystals are maximally rigid on any compact.