Let $Z_1, \cdots, Z_n$ denote the eigenvalues of the product $\prod_{j=1}^{k_n} \boldsymbol{A}_j$, where $\{\boldsymbol{A}_j\}_{1 \le j \le k_n}$ are independent $n\times n$ complex Ginibre matrices. Define $\alpha = \lim\limits_{n \to \infty} \frac{n}{k_n}$. We prove that $X_n,$ a suitably rescaled version of $\max_{1 \le j \le n} |Z_j|^2,$ converges weakly as follows: to a non-trivial distribution $\Phi_\alpha$ for $\alpha \in (0, +\infty)$, to the Gumbel distribution when $\alpha = +\infty$, 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.

翻译：令 $Z_1, \cdots, Z_n$ 表示乘积 $\prod_{j=1}^{k_n} \boldsymbol{A}_j$ 的特征值，其中 $\{\boldsymbol{A}_j\}_{1 \le j \le k_n}$ 是独立的 $n\times n$ 复 Ginibre 矩阵。定义 $\alpha = \lim\limits_{n \to \infty} \frac{n}{k_n}$。我们证明，经过适当重标度的 $X_n = \max_{1 \le j \le n} |Z_j|^2$ 弱收敛如下：当 $\alpha \in (0, +\infty)$ 时收敛于非平凡分布 $\Phi_\alpha$；当 $\alpha = +\infty$ 时收敛于 Gumbel 分布；当 $\alpha = 0$ 时收敛于标准正态分布。该结果揭示了在 $\alpha$ 的边界处存在相变。此外，我们建立了每种情形下的精确收敛速率。