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A conjecture of Aldous and Diaconis (1985) asserts, for $k\\gg\\log|G|$, that the random walk on this graph exhibits cutoff.\n  When $\\log k \\lesssim \\log\\log|G|$ (ie $k = (\\log |G|)^{\\mathcal O(1)}$), the only example of a non-Abelian group for which cutoff has been established is the dihedral group. We establish cutoff (as $p\\to infty$) for the group of $d \\times d$ unit upper triangular matrices with integer entries modu"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1911.02974","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2019-11-07T17:22:00Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"af6c1c21e3873c7973f36417277cfdb629ccb0c9111bb0bc0c3357f1c0745e15","abstract_canon_sha256":"4b1ddacfe3d9546309013928b810a3c64fafafa045c752ac2ab987f4cdf57829"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:12:39.997100Z","signature_b64":"LmfP3aIRQmlL6qB71PY1jTO+vu4wlhJ+90HrVdRwPQY+cDaxoMzr9y6PDsxFZgzvRLZFiXvn9j96n2mToQ5sAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02f2139f464e16c6910ca2f18be6e58e4e7d0c36d0ecf99964ed78a6059d7889","last_reissued_at":"2026-07-05T02:12:39.996671Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:12:39.996671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cutoff for Random Walks on Upper Triangular Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.PR","authors_text":"Jonathan Hermon, Sam Olesker-Taylor","submitted_at":"2019-11-07T17:22:00Z","abstract_excerpt":"Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \\ll \\log k \\ll \\log |G|$ (ie $1 \\ll k = |G|^{o(1)}$). A conjecture of Aldous and Diaconis (1985) asserts, for $k\\gg\\log|G|$, that the random walk on this graph exhibits cutoff.\n  When $\\log k \\lesssim \\log\\log|G|$ (ie $k = (\\log |G|)^{\\mathcal O(1)}$), the only example of a non-Abelian group for which cutoff has been established is the dihedral group. We establish cutoff (as $p\\to infty$) for the group of $d \\times d$ unit upper triangular matrices with integer entries modu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.02974","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1911.02974/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1911.02974","created_at":"2026-07-05T02:12:39.996733+00:00"},{"alias_kind":"arxiv_version","alias_value":"1911.02974v2","created_at":"2026-07-05T02:12:39.996733+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.02974","created_at":"2026-07-05T02:12:39.996733+00:00"},{"alias_kind":"pith_short_12","alias_value":"ALZBHH2GJYLM","created_at":"2026-07-05T02:12:39.996733+00:00"},{"alias_kind":"pith_short_16","alias_value":"ALZBHH2GJYLMNEIM","created_at":"2026-07-05T02:12:39.996733+00:00"},{"alias_kind":"pith_short_8","alias_value":"ALZBHH2G","created_at":"2026-07-05T02:12:39.996733+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2401.03937","citing_title":"Cutoff for mixtures of permuted Markov chains: reversible case","ref_index":38,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ","json":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ.json","graph_json":"https://pith.science/api/pith-number/ALZBHH2GJYLMNEIMULYYXZXFRZ/graph.json","events_json":"https://pith.science/api/pith-number/ALZBHH2GJYLMNEIMULYYXZXFRZ/events.json","paper":"https://pith.science/paper/ALZBHH2G"},"agent_actions":{"view_html":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ","download_json":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ.json","view_paper":"https://pith.science/paper/ALZBHH2G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1911.02974&json=true","fetch_graph":"https://pith.science/api/pith-number/ALZBHH2GJYLMNEIMULYYXZXFRZ/graph.json","fetch_events":"https://pith.science/api/pith-number/ALZBHH2GJYLMNEIMULYYXZXFRZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ/action/storage_attestation","attest_author":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ/action/author_attestation","sign_citation":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ/action/citation_signature","submit_replication":"https://pith.science/pith/ALZBHH2GJYLMNEIMULYYXZXFRZ/action/replication_record"}},"created_at":"2026-07-05T02:12:39.996733+00:00","updated_at":"2026-07-05T02:12:39.996733+00:00"}