LOCAL LAWS FOR MULTIPLICATION OF RANDOM MATRICES

• 成果类型: Article
• 署名作者: Ding, Xiucai; Ji, Hong Chang
• 署名单位: University of California System; University of California Davis; Institute of Science & Technology - Austria
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1882
• 发表日期: 2023
• 页码: 2981-3009
• 关键词: sample covariance matrices; tracy-widom distribution; Spectral Distribution; LARGEST EIGENVALUE; SUBORDINATION; eigenvectors; UNIVERSALITY; convolution; REGULARITY; support

摘要

Consider the random matrix model A1/2UB U*A1/2, where A and B are two N x N deterministic matrices and U is either an N x N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201-220), the limiting empirical spectral distribu-tion (ESD) of the above model is given by the free multiplicative convolution of the limiting ESDs of A and B, denoted as & mu;& alpha; & REG; & mu;⠃, where & mu;& alpha; and & mu;⠃ are the limiting ESDs of A and B, respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of & mu;A & REG; & mu;B, where & mu;A and & mu;B are the ESDs of A and B, respectively and the associated subordination functions have a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of A, B and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the addi-tive model A + UBU* in (J. Funct. Anal. 271 (2016) 672-719; Comm. Math. Phys. 349 (2017) 947-990; Adv. Math. 319 (2017) 251-291; J. Funct. Anal. 279 (2020) 108639), and our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020) 108639) for the multiplicative model.