Rigidity and amesoscopic central limit theorem for Dyson Brownian motion for general β and potentials

• 成果类型: Article
• 署名作者: Huang, Jiaoyang; Landon, Benjamin
• 署名单位: Harvard University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0889-y
• 发表日期: 2019
• 页码: 209-253
• 关键词: statistical theory; energy levels; eigenvalue statistics; global fluctuations; UNIVERSALITY; matrices

摘要

We study Dyson Brownian motion with general potential V and for general beta >= 1. For short times t = o(1) and under suitable conditions on V we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale eta(*) << t << 1, we prove a mesoscopic central limit theorem for linear statistics at all scales eta with N-1 << eta << t.