{"paper":{"title":"Sharpness of Bernoulli percolation via couplings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Hugo Vanneuville","submitted_at":"2022-01-20T15:27:54Z","abstract_excerpt":"In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\\mathbb{Z}^d$). We prove the following estimate, where $\\theta_n(p)$ is the probability that there is a path of $p$-open edges from $0$ to the sphere of radius $n$: \\[ \\forall p\\in [0,1],\\forall m,n \\ge 1, \\quad \\theta_{2n} (p-2\\theta_m(p))\\le C\\frac{\\theta_n(p)}{2^{n/m}}. \\] This result implies that $\\theta_n(p)$ decays exponentially fast in the subcritical phase. It also implies the mean-field lower bound in the supercritical phase. We thus provide a new proof of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.08223","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/2201.08223/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"}