Precise convergence rate of spectral radius of product of complex Ginibre
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911585840136192 |
|---|---|
| author | Ma, Yutao Meng, Xujia |
| author_facet | Ma, Yutao Meng, Xujia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Precise convergence rate of spectral radius of product of complex Ginibre Ma, Yutao Meng, Xujia Probability Statistics Theory 60G70, 60B20, 15B52 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. |
| title | Precise convergence rate of spectral radius of product of complex Ginibre |
| topic | Probability Statistics Theory 60G70, 60B20, 15B52 |
| url | https://arxiv.org/abs/2510.07942 |