标题： 复Ginibre矩阵乘积谱半径的精确收敛速率 显示英文标题

标题： Precise convergence rate of spectral radius of product of complex Ginibre

摘要： 设$Z_1, \cdots, Z_n$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$n\times n$复数Ginibre矩阵。 定义$\alpha = \lim\limits_{n \to \infty} \frac{n}{k_n}$。 我们证明了$X_n,$是$\max_{1 \le j \le n} |Z_j|^2,$的一个适当缩放版本，其弱收敛如下：对于$\alpha \in (0, +\infty)$为非退化的分布$\Phi_\alpha$，当$\alpha = +\infty$时为 Gumbel 分布，当$\alpha = 0$时为标准正态分布。 这一结果揭示了$\alpha$边界处的相变。 此外，我们在每个区域中建立了收敛的精确速率。 显示英文摘要

摘要： 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.