EXTREMAL STATISTICS OF QUADRATIC FORMS OF GOE/GUE EIGENVECTORS

• 成果类型: Article
• 署名作者: Erdos, Laszlo; McKenna, Benjamin
• 署名单位: Institute of Science & Technology - Austria
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP2000
• 发表日期: 2024
• 页码: 1623-1662
• 关键词: level-spacing distributions; local semicircle law; RANDOM MATRICES; edge universality; eigenvalues; RIGIDITY; maximum; ENTRIES

摘要

We consider quadratic forms of deterministic matrices A evaluated at the random eigenvectors of a large N x N GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than root N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of A. Our result also naturally applies to the eigenvectors of any invariant ensemble.