{"paper":{"title":"Mixing on $k$ Columns of the Transvection Walk","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Aaron Smith, Natesh Pillai","submitted_at":"2026-05-26T16:03:23Z","abstract_excerpt":"In Diaconis and Saloff-Coste (1996), the authors introduced the simple ``transvection\" walk on $\\mathrm{GL}_n(\\mathbb F_2)$: at each step, choose two distinct rows and add one to the other. In Ben-Hamou (2025), the author recently proved that this walk has mixing time $O(n^2\\log n)$. Inspired by applications in cryptography (see Sotiraki (2016)), Ben-Hamou and Peres (2018) conjectured that the first $k$ columns of this walk mixed in $O(nk \\log(n))$ steps. Our main result is a proof of this conjecture uniformly in $n$ and $k.$\n  Our proof is based on a local-to-global entropy estimate, in the s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.27212","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/2605.27212/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"}