Precise convergence rate of spectral radius of product of complex Ginibre

Let $Z_1,\cdots ,Z_n$ denote the eigenvalues of the product ${\prod }_{j=1}^{k_n}{\mathit{A}}_j$, where $\{{\mathit{A}}_j{\}}_{1\le j\le k_n}$ are independent $n\times n$ complex Ginibre matrices. Define $\alpha =\underset{n\to \mathrm{\infty }}{\lim }\frac{n}{k_n}$. We prove that $X_n,$ a suitably rescaled version of ${\max }_{1\le j\le n}\mathrm{\mid }{Z}_j{\mathrm{\mid }}^2,$ converges weakly as follows: to a non-trivial distribution ${\mathrm{\Phi }}_{\alpha }$ for $\alpha \in (0,+\mathrm{\infty })$, to the Gumbel distribution when $\alpha =+\mathrm{\infty }$, and to the standard normal distribution when $\alpha =0$. This result reveals a phase transition at the boundaries of $\alpha$. Furthermore, we establish the exact rates of convergence in each regime.