Precise convergence rate of spectral radius of product of complex Ginibre

Abstract

Let $Z_1,\cdots ,Z_n$ denote the eigenvalues of the product ${\prod }_{j=1}^{k_n}{\mathbf{A}}_j$, where $\{{\mathbf{A}}_j{\}}_{1\le j\le k_n}$ are independent $n\times n$ complex Ginibre matrices. Define $\alpha =\underset{n\to \infty }{\lim }\frac{n}{k_n}$. 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.

This version makes substantial improvements over the previous one, including in the title, abstract, and content. We would therefore prefer to announce it as a completely new submission