{"paper":{"title":"Mixing times of Langevin dynamics for spiked matrix models","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Langevin dynamics for large-signal spiked matrices mix in O(log N) from uniform spherical starts even below the critical temperature.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Curtis Grant, Reza Gheissari, Tianmin Yu","submitted_at":"2026-04-21T21:36:21Z","abstract_excerpt":"We investigate the Langevin dynamics for Wigner matrices with a spherical spike, in the regime where the signal-to-noise ratio $\\theta$ is large, but order one. For large, order-$1$, signal-to-noise, the (worst-case) mixing time undergoes a sharp transition around the critical inverse temperature $\\beta_c(\\theta) = \\frac{1}{\\theta}$. Namely, if $\\beta = \\alpha/\\theta$, and $\\alpha<1$ then at large $\\theta$ the mixing time is $O(\\log N)$, and if $\\alpha>1$ it is exponential in $N$. We show that initialized from the uniform-at-random spherical prior, however, the mixing time in the low-temperatu"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Initialized from the uniform-at-random spherical prior, the mixing time in the low-temperature α>1 regime is O(log N). The exact exponential rate of the (worst-case initialization) mixing time for low temperatures is given by the difference of the free energies of the spiked and null models.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis is performed in the regime where the signal-to-noise ratio θ is large but remains order one; the fast-mixing claim requires the initialization to be symmetric with respect to the top eigenvector of the spiked matrix.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For spiked Wigner matrices, Langevin dynamics mixes in O(log N) time from uniform or top-eigenvector-symmetric starts below the critical inverse temperature 1/θ, while worst-case mixing is exponential in N with rate equal to the free-energy difference between spiked and null models.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Langevin dynamics for large-signal spiked matrices mix in O(log N) from uniform spherical starts even below the critical temperature.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"398f60aac61ce0db9bd07201b13944de4682fbac944d41d22acf1d4eb6e9f30f"},"source":{"id":"2604.20008","kind":"arxiv","version":2},"verdict":{"id":"704584ca-020c-4ed8-a2f1-df676bd17f3c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T01:01:16.493406Z","strongest_claim":"Initialized from the uniform-at-random spherical prior, the mixing time in the low-temperature α>1 regime is O(log N). The exact exponential rate of the (worst-case initialization) mixing time for low temperatures is given by the difference of the free energies of the spiked and null models.","one_line_summary":"For spiked Wigner matrices, Langevin dynamics mixes in O(log N) time from uniform or top-eigenvector-symmetric starts below the critical inverse temperature 1/θ, while worst-case mixing is exponential in N with rate equal to the free-energy difference between spiked and null models.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis is performed in the regime where the signal-to-noise ratio θ is large but remains order one; the fast-mixing claim requires the initialization to be symmetric with respect to the top eigenvector of the spiked matrix.","pith_extraction_headline":"Langevin dynamics for large-signal spiked matrices mix in O(log N) from uniform spherical starts even below the critical temperature."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20008/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T15:38:45.753165Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T02:24:28.605892Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"341c4f40c8be1a6f60ad6a35cfd0ad3271fdbf40b2254daf3fcc6ab3c49f09f0"},"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"}