{"paper":{"title":"Rigidity of random stationary measures and applications to point processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"MAP5 - UMR 8145), Rapha\\\"el Lachi\\`eze-Rey (DYOGENE","submitted_at":"2024-09-27T07:56:09Z","abstract_excerpt":"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 measur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.18519","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.18519/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}