## Abstract

Consider the product of m independent n-by-n Ginibre matrices and their inverses, where m=p+q, 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 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, 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.

Original language | English (US) |
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Journal | Journal of Theoretical Probability |

DOIs | |

State | Accepted/In press - 2024 |

### Bibliographical note

Publisher Copyright:© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

## Keywords

- 60B20
- 60F05
- 60G70
- Eigenvalue
- Ginibre matrix
- Product matrix
- Spectral radius