Local spectral statistics of the addition of random matrices

• 成果类型: Article
• 署名作者: Che, Ziliang; Landon, Benjamin
• 署名单位: Harvard University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00932-2
• 发表日期: 2019
• 页码: 579-654
• 关键词: bulk universality; eigenvalue statistics; generalized wigner; CONVERGENCE; SUBORDINATION; SUM; LAW

摘要

We consider the local statistics of H = V * XV + U * YU where V and U are independent Haar-distributed unitary matrices, and X and Y are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size N -> infinity under mild assumptions on X and certain rigidity assumptions on Y (the latter being an assumption on the convergence of the eigenvalues of Y to the quantiles of its limiting spectral measure which we assume to have a density). Our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to Dyson Brownian motion. Our method also applies to the case when V and U are drawn from the orthogonal group. Our proof relies on the local law for H proved in Bao et al. (Commun Math Phys 349(3):947-990, 2017; J Funct Anal 271(3):672-719, 2016; Adv Math 319:251-291, 2017) as well as the DBM convergence results of Landon and Yau (Commun Math Phys 355(3):949-1000, 2017) and Landon et al. (Adv Math 346:1137-1332, 2019).