{"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. This flow is stationary with respect to the distribution \\[\n  \\rho^{\\beta}_N(\\lambda) = \\frac{1}{Z^{\\beta}_N} e^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02700","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":""},"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"}