Eigenvector distribution in the critical regime of BBP transition

• 成果类型: Article
• 署名作者: Bao, Zhigang; Wang, Dong
• 署名单位: Hong Kong University of Science & Technology; National University of Singapore
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-021-01062-4
• 发表日期: 2022
• 页码: 399-479
• 关键词: finite rank deformations; central limit-theorems; LARGEST EIGENVALUE; asymptotic solutions; external source; matrices; fluctuations; perturbations; airy

摘要

In this paper, we study the random matrix model of Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source. We will focus on the critical regime of the Baik-Ben Arous-Peche (BBP) phase transition and establish the distribution of the eigenvectors associated with the leading eigenvalues. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition [6]. The derivation of the distribution makes use of the recently re-discovered eigenvector-eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source.