{"paper":{"title":"A Poincar\\'e Inequality and Exponential Decay for the Elephant Random Walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Cristian Favio Coletti, Ioannis Papageorgiou, Rafael de Mattos Grisi","submitted_at":"2026-06-07T23:56:13Z","abstract_excerpt":"We study the long-time behaviour of a coninuous time one-dimensional elephant random walk with an absorbing boundary. By analyzing the associated evolution equation, we identify a proper limiting operator and establish a Poincar\\'e inequality with spectral gap of order $N^{-2}$. As a consequence, we obtain matching exponential upper and lower bounds for the survival probability, showing that it decays at rate $e^{-ct/N^2}$. The proof relies on a decomposition of the generator into a limiting operator and a time-dependent perturbation, together with spectral estimates."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08884","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/2606.08884/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"}