Limiting Spectral Radii for Products of Ginibre Matrices and Their Inverses

Consider the product of m independent n-by-n Ginibre matrices and their inverses, where m=p+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=p+q$$\end{document}, p is the number of Ginibre matrices, and q is the number of inverses of Ginibre matrices. The maximum absolute value of the eigenvalues of the product matrices is known as the spectral radius. In this paper, we explore the limiting spectral radii of the product matrices as n tends to infinity and m varies with n. Specifically, when q >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 1$$\end{document} is a fixed integer, we demonstrate that the limiting spectral radii display a transition phenomenon when the limit of p/n changes from zero to infinity. When q=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=0$$\end{document}, the limiting spectral radii for Ginibre matrices have been obtained by Jiang and Qi [J Theor Probab 30: 326-364, 2017]. When q diverges to infinity as n approaches infinity, we prove that the logarithmic spectral radii exhibit a normal limit, which reduces to the limiting distribution for spectral radii for the spherical ensemble obtained by Chang et al. [J Math Anal Appl 461: 1165-1176, 2018] when p=q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=q$$\end{document}.