{"paper":{"title":"The Large-$N$ Limits of Brownian Motions on $\\mathbb{GL}_N$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Todd Kemp","submitted_at":"2013-06-25T17:01:12Z","abstract_excerpt":"We introduce a two-parameter family of diffusion processes $(B_{r,s}^N(t))_{t\\ge 0}$, $r,s>0$, on the general linear group $\\mathbb{GL}_N$ that are Brownian motions with respect to certain natural metrics on the group. At the same time, we introduce a two-parameter family of free It\\^o processes $(b_{r,s}(t))_{t\\ge 0}$ in a faithful, tracial $W^\\ast$-probability space, and we prove that the full process $(B^N_{r,s}(t))_{t\\ge 0}$ converges to $(b_{r,s}(t))_{t\\ge 0}$ in noncommutative distribution as $N\\to\\infty$ for each $r,s>0$. The processes $(b_{r,s}(t))_{t\\ge 0}$ interpolate between the fre"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.6033","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"}