{"paper":{"title":"Hit-and-Run Mixes as Fast as the Ball Walk","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cs.DS","authors_text":"Ruizhe Zhang","submitted_at":"2026-08-13T17:15:31Z","abstract_excerpt":"Let $K\\subset\\mathbb{R}^n$ be an isotropic convex body. We prove that the hit-and-run walk, started from any $M$-warm distribution, reaches total-variation distance $\\varepsilon$ from the uniform distribution on $K$ in $O\\!\\left(n^2\\psi_n^{-2}\\log^3(M/\\varepsilon)\\right)$ steps, where $\\psi_n^{-1}$ is the Kannan-Lov\\'asz-Simonovits (KLS) constant. Up to logarithmic factors, this matches the best-known warm-start mixing time for the ball walk. Chen and Eldan [Discrete Comput. Geom. 2026] obtained the same $n^2\\psi_n^{-2}$ dependence for hit-and-run, but with polynomial dependence on $M/\\varepsi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.13487","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/2608.13487/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"}