Local law and rigidity for unitary Brownian motion

• 成果类型: Article
• 署名作者: Adhikari, Arka; Landon, Benjamin
• 署名单位: Stanford University; University of Toronto
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01230-8
• 发表日期: 2023
• 页码: 753-815
• 关键词: random matrices universality; central-limit-theorem; semicircle law; eigenvalues; delocalization; eigenvectors; statistics

摘要

We establish high probability estimates on the eigenvalue locations of Brownian motion on the N-dimensional unitary group, as well as estimates on the number of eigenvalues lying in any interval on the unit circle. These estimates are optimal up to arbitrarily small polynomial factors in N. Our results hold at the spectral edges (showing that the extremal eigenvalues are within O(N-2/3+) of the edges of the limiting spectral measure), in the spectral bulk, as well as for times near 4 at which point the limiting spectral measure forms a cusp. Our methods are dynamical and are based on analyzing the evolution of the Cauchy transform of the empirical spectral measure along the characteristics of the PDE satisfied by the limiting spectral measure, that of the free unitary Brownian motion.