{"paper":{"title":"A phase transition in the Bakry-\\'Emery gradient estimate for Dyson Brownian motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.PR","authors_text":"Kenshiro Tashiro, Kohei Suzuki","submitted_at":"2025-06-04T20:17:36Z","abstract_excerpt":"In this paper, we find a gap between the lower bound of the Bakry-\\'Emery $N$-Ricci tensor ${\\rm Ric}_N$ and the Bakry-\\'Emery gradient estimate ${\\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<\\beta<1$. Namely, we prove that, for the weighted space $(\\mathbb R^n, w_\\beta)$ with $w_\\beta=\\prod_{i<j}^n |x_i-x_j|^\\beta$ and any $N\\in[n+\\frac{\\beta}{2}n(n-1),+\\infty]$,\n  $\\beta \\ge 1 \\implies {\\rm Ric}_N \\ge 0 \\ \\& \\ {\\sf BE}(0,N)$ hold;\n  $0 < \\beta < 1 \\implies {\\rm Ric}_N \\ge 0$ holds while ${\\sf BE}(0,N)$ does not, which shows"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.04424","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.04424/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"}