Rigidity of random stationary measures and applications to point processes
Raphaël Lachièze-Rey
TL;DR
This work develops a unifying spectral framework linking rigidity of random stationary measures to the near-origin behavior of the structure factor $\\mathscr s$. By leveraging Schwartz’s Paley–Wiener theorem, it derives both sufficient and (under structural assumptions) necessary conditions for $k$-rigidity in continuous and discrete settings, revealing that a zero/pole structure of $\\mathscr s^{-1}$ at the origin governs how many moments of a localized statistic can be recovered from outside observations. The results yield concrete criteria for determinantal point processes, provide insight into the rigidity of Gibbs-type models, and establish maximal rigidity for random stationary quasicrystals, while also constructing models with arbitrarily high rigidity and clarifying limitations in higher dimensions. Overall, the paper ties hyperuniformity, spectral decay, and analytic-type conditions to precise rigidity phenomena across a broad class of random measures and point processes, offering tools to assess completeness and phase behavior in complex systems.
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 $κ$, $k$-rigidity is equivalent to $(1- \widehat {κ^{2}})^{-1}$ 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.