Edge universality of sparse Erdős-Rényi digraphs

• 成果类型: Article; Early Access
• 署名作者: He, Yukun
• 署名单位: Fudan University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01423-3
• 发表日期: 2025
• 关键词: circular law; RANDOM MATRICES; extremal eigenvalues; spectral statistics

摘要

Let A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} be the adjacency matrix of the Erd & odblac;s-R & eacute;nyi directed graph G(N,p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {G}}(N,p)$$\end{document}. We denote the eigenvalues of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} by lambda 1A,...,lambda NA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1<^>{{\mathcal {A}}},...,\lambda <^>{\mathcal {A}}_N$$\end{document}, with |lambda 1A|=maxi|lambda iA|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\lambda _1<^>{{\mathcal {A}}}|=\max _i|\lambda _i<^>{{\mathcal {A}}}|$$\end{document}. For N-1+o(1)<= p <= 1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N<^>{-1+o(1)}\leqslant p\leqslant 1/2$$\end{document}, we show that maxi=2,3,...,N|lambda iANp(1-p)|=1+O(N-1/2+o(1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \max _{i=2,3,...,N} \bigg |\frac{\lambda _i<^>{{\mathcal {A}}}}{\sqrt{Np(1-p)}}\bigg | =1+O(N<^>{-1/2+o(1)}) $$\end{document}with very high probability. In addition, we prove that near the unit circle, the local eigenvalue statistics of A/Np(1-p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}/\sqrt{Np(1-p)}$$\end{document} coincide with those of the real Ginibre ensemble. As a by-product, we also show that all non-trivial eigenvectors of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {A}}$$\end{document} are completely delocalized. For Hermitian random matrices, it is known that the edge statistics are sensitive to the sparsity: in the very sparse regime, one needs to remove many noise random variables (which affect both the mean and the fluctuation) to recover the Tracy-Widom distribution (Erd & odblac;s, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of erd & odblac;s-r & eacute;nyi graphs i: local semicircle law. Ann. Prob. 41, 2279-2375 (2013)), (Erd & odblac;s, L., Knowles, A., Yau, H.-T. , Yin, J.: Spectral statistics of erd & odblac;s-r & eacute;nyi graphs ii: eigenvalue spacing and the extreme eigenvalues. Comm. Math. Phys. 314, 587-640 (2012)), (Lee, J.O., Schnelli, K.: Local law and tracy-widom limit for sparse random matrices. Prob. Theor. Rel. Fields 171, 543-616 (2018)), (Huang, J., Landon, B., Yau, H.-T.: Transition from tracy-widom to gaussian fluctuations of extremal eigenvalues of sparse erd & odblac;s-r & eacute;nyi graphs. Ann. Prob. 48, 916-962 (2020)), (He, Y., Knowles, A.: Fluctuations of extreme eigenvalues of sparse erd & odblac;s-r & eacute;nyi graphs. Prob. Theor. Rel. Fields 180, 985-1056 (2021)), (Lee, J.: Higher order fluctuations of extremal eigenvalues of sparse random matrices, Preprint arXiv:2108.11634), (Huang, J., Yau, H.T.: Edge universality of sparse random matrices, Preprint arXiv: 2206.06580). Our results imply that, compared to their analogues in the Hermitian case, the edge statistics of non-Hermitian sparse random matrices are more robust.