Precise convergence rate of spectral radius of product of complex Ginibre
Yutao Ma, Xujia Meng
TL;DR
The paper derives precise convergence rates for the extreme eigenvalue behavior of the product of independent complex Ginibre matrices, revealing a phase transition governed by $\alpha=\lim_{n\to\infty} n/k_n$. By a Kostlan-type decoupling and Edgeworth expansions, the authors show that the rescaled maximum eigenvalue converges to a limiting distribution $\Phi_\alpha$ for $\alpha\in(0,\infty)$, to the Gumbel law for $\alpha=+\infty$, and to the normal law for $\alpha=0$, with detailed Berry–Esseen-type rates that depend on the asymptotics of $k_n$ relative to $n$. They provide sharp bounds for the transition function $\Phi_\alpha$, quantify how it interpolates between Gaussian and Gumbel limits as $\alpha$ varies, and supply explicit constants for the convergence rates, including Wasserstein-distance analogues. The results extend prior work on spectral-radius limits and illuminate the precise manner in which phase transitions manifest in non-Hermitian random matrix products. The techniques combine decoupling, Edgeworth expansions, interval partitioning, and careful tail analysis to achieve uniform, fine-grained rates across regimes.
Abstract
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 $α= \lim\limits_{n \to \infty} \frac{n}{k_n}$. 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 $Φ_α$ for $α\in (0, +\infty)$, to the Gumbel distribution when $α= +\infty$, and to the standard normal distribution when $α= 0$. This result reveals a phase transition at the boundaries of $α$. Furthermore, we establish the exact rates of convergence in each regime.