Dyson Brownianmotion for general β and potential at the edge

• 成果类型: Article
• 署名作者: Adhikari, Arka; Huang, Jiaoyang
• 署名单位: Harvard University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00992-9
• 发表日期: 2020
• 页码: 893-950
• 关键词: random matrices universality; fixed-energy universality; eigenvalue statistics; bulk universality; characteristic vectors; bordered matrices; wigner matrices; semicircle law; RIGIDITY; motion

摘要

In this paper, we compare the solutions of the Dyson Brownian motion for general beta and potential V and the associated McKean-Vlasov equation near the edge. Under suitable conditions on the initial data and the potential V, we obtain optimal rigidity estimates of particle locations near the edge after a short time t = o(1). Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge, which, as far as we know, has been done for the first time in this paper. Additionally, combining our results with Landon and Yau (Edge statistics of Dyson Brownian motion. arXiv:1712.03881, 2017), we give a proof of the local ergodicity of the Dyson Brownian motion for general beta and potential at the edge, i.e., we showthe distribution of extreme particles converges to the Tracy-Widom beta distribution in a short time.