{"paper":{"title":"Minimum weight disk triangulations and fillings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Eyal Lubetzky, Itai Benjamini, Yuval Peled","submitted_at":"2019-11-06T19:00:00Z","abstract_excerpt":"We study the minimum total weight of a disk triangulation using vertices out of $\\{1,\\ldots,n\\}$, where the boundary is the triangle $(123)$ and the $\\binom{n}3$ triangles have independent weights, e.g. $\\mathrm{Exp}(1)$ or $\\mathrm{U}(0,1)$.\n  We show that for explicit constants $c_1,c_2>0$, this minimum is $c_1 \\frac{\\log n}{\\sqrt n} + c_2 \\frac{\\log\\log n}{\\sqrt n} + \\frac{Y_n}{\\sqrt n}$ where the random variable $Y_n$ is tight, and it is attained by a triangulation that consists of $\\frac14\\log n + O_P(\\sqrt{\\log n}) $ vertices.\n  Moreover, for disk triangulations that are canonical, in th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.02569","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/1911.02569/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"}