Phase transition of eigenvalues in deformed Ginibre ensembles I: GinUE

• 成果类型: Article; Early Access
• 署名作者: Liu, Dang-Zheng; Zhang, Lu
• 署名单位: Chinese Academy of Sciences; University of Science & Technology of China, CAS; Chinese Academy of Sciences; University of Science & Technology of China, CAS
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01318-9
• 发表日期: 2024
• 关键词: random matrices; circular law; UNIVERSALITY; outliers

摘要

Consider a random matrix of size N as an additive deformation of the complex Ginibre ensemble under a deterministic matrix X0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} with a finite rank, independent of N. When some eigenvalues of X0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of X0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document}. These findings are largely due to Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 165:313-363, 2016), Bordenave and Capitaine (Comm. Pure Appl. Math. 69:2131-2194, 2016), and Tao (Probab. Theory Relat. Fields 155:231-263, 2013). Yet when there is an eigenvalue of X0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_0$$\end{document} on the edge of the unit disk, we prove that local eigenvalue statistics at the same spectral edge form a new class of determinantal point processes, for which correlation kernels only depend on geometric multiplicity of eigenvalue and are characterized in terms of the iterated erfc functions. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory.