2024/09/27 by Raphaël Lachièze-Rey, Lachièze-Rey, Raphaël · 2 citations
Mathematics · #FOS: Mathematics #Morphological variations and asymmetry #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2409.18519
openalex publication_date 2024/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The \it number rigidity of a stationary point process P entails that for a bounded set A the knowledge of P on Ac a.s. determines P(A); the k-order rigidity means the moments of P1A up to order k can be recovered. We show that k-rigidity occurs if the continuous component \mathscrs of 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 L2 wide sense stationary measure on ℝd or ℤd. In the continuous setting, these local conditions are also necessary if \mathscrs 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 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 κ2)-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.