{"paper":{"title":"Small ball probabilities for the passage time in planar first-passage percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dor Elboim","submitted_at":"2024-06-16T15:09:21Z","abstract_excerpt":"We study planar first-passage percolation with independent weights whose common distribution is supported in $(0,\\infty)$ and is absolutely continuous with respect to Lebesgue measure. We prove that the passage time from $x$ to $y$ denoted by $T(x,y)$ satisfies $$\\max _{a\\ge 0} \\mathbb P \\big( T(x,y)\\in [a,a+1] \\big) \\le \\frac{C}{\\sqrt{\\log \\|x-y\\|}},$$ answering a question posed by Ahlberg and de la Riva. This estimate recovers earlier results on the fluctuations of the passage time by Newman--Piza, Pemantle--Peres, and Chatterjee."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2406.10971","kind":"arxiv","version":2},"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/2406.10971/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"}