{"paper":{"title":"Average Gromov hyperbolicity and the Parisi ansatz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MG","math.MP"],"primary_cat":"math.PR","authors_text":"Leila Sloman, Sourav Chatterjee","submitted_at":"2019-07-06T23:20:59Z","abstract_excerpt":"Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a \"worst-case\" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an \"average-case\" version of Gromov hyperbolicity, which detects whether the \"most of the space\", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03203","kind":"arxiv","version":6},"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/1907.03203/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"}