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This is the stochastic gradient flow on $\\mathbb{R}^n$ given by, for all $1 \\leq i \\leq n,$ \\[\n  d\\lambda_{i,t} = \\sqrt{\\frac{2}{\\beta}}dZ_{i,t} - \\biggl( \\frac{V'(\\lambda_i)}{2} - \\sum_{j: j \\neq i} \\frac{1}{\\lambda_i - \\lambda_j} \\biggr)\\,dt \\] where $V$ is a constraining potential and $\\left\\{ Z_{i,t} \\right\\}_1^n$ are independent standard Brownian motions. This flow is stationary with respect to the distribution \\[\n  \\rho^{\\beta}_N(\\lambda) = \\frac{1}{Z^{\\beta}_N} e^{"},"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":"1707.02700","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-10T05:31:44Z","cross_cats_sorted":[],"title_canon_sha256":"1241403609381d6ca20872a5019994103c6e8eb26a4506e2787a08bda97c524a","abstract_canon_sha256":"484d519b52f976f50409cbb6dba3f55e35b4f8d7d40ec8add121d36db083cadb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:36.096578Z","signature_b64":"EGyIfx7H4Mi6GtIgpvON1wG/PAV3Lc5Jdw0qKvw304bFdZpKzQJMvnVWh7b9UCTAxP9Apd+SzctNiCkzcyShCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca7482e551d7a4831b6791ca89850970953684ab5a80fe48b3977144fce21ebc","last_reissued_at":"2026-05-18T00:40:36.095895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:36.095895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tridiagonal Models for Dyson Brownian Motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Diane Holcomb, Elliot Paquette","submitted_at":"2017-07-10T05:31:44Z","abstract_excerpt":"In this paper, we consider tridiagonal matrices the eigenvalues of which evolve according to $\\beta$-Dyson Brownian motion. This is the stochastic gradient flow on $\\mathbb{R}^n$ given by, for all $1 \\leq i \\leq n,$ \\[\n  d\\lambda_{i,t} = \\sqrt{\\frac{2}{\\beta}}dZ_{i,t} - \\biggl( \\frac{V'(\\lambda_i)}{2} - \\sum_{j: j \\neq i} \\frac{1}{\\lambda_i - \\lambda_j} \\biggr)\\,dt \\] where $V$ is a constraining potential and $\\left\\{ Z_{i,t} \\right\\}_1^n$ are independent standard Brownian motions. 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