{"paper":{"title":"Gravitational allocation for uniform points on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alex Zhai, Nina Holden, Yuval Peres","submitted_at":"2017-04-26T17:46:06Z","abstract_excerpt":"Given a collection $\\mathcal L$ of $n$ points on a sphere $\\mathbf{S}^2_n$ of surface area $n$, a fair allocation is a partition of the sphere into $n$ parts each of area $1$, and each associated with a distinct point of $\\mathcal L$. We show that if the $n$ points are chosen uniformly at random and the partition is defined by considering the gravitational field defined by the $n$ points, then the expected distance between a point on the sphere and the associated point of $\\mathcal L$ is $O(\\sqrt{\\log n})$. We use our result to define a matching between two collections of $n$ independent and u"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.08238","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":""},"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"}