{"paper":{"title":"Localization and unique continuation for the Anderson-Bernoulli model with long-range hopping on $\\mathbb{Z}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.DS","math.MP","math.PR"],"primary_cat":"math.SP","authors_text":"Shihe Liu, Yunfeng Shi, Zhifei Zhang","submitted_at":"2026-07-03T16:33:01Z","abstract_excerpt":"In this paper, we study Anderson localization near the spectral edge for the Anderson-Bernoulli model on $\\mathbb{Z}$ with long-range hopping. When the hopping has a rational Laurent symbol, a quantitative version of the unique continuation principle can be proved, and localization occurs. For the unique continuation in the general case, we give some counterexamples and prove a weaker result for hopping that decays faster than exponential rate. To the best of our knowledge, this is the first localization result for the long-range Anderson model with pure Bernoulli potentials."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03472","kind":"arxiv","version":1},"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/2607.03472/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"}