Universality of the convergence rate for spectral radius of complex IID random matrices
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| Format: | Preprint |
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2025
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| _version_ | 1866909642263625728 |
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| author | Hu, Xinchen Ma, Yutao |
| author_facet | Hu, Xinchen Ma, Yutao |
| contents | Let $X$ be an $n\times n$ matrix with independent and identically distributed entries $x_{ij} \stackrel{\text { d }}{=} n^{-1 / 2} x$ for some complex random variable $x$ of mean zero and variance one. Let $\{σ_i\}_{1\le i\le n}$ be the eigenvalues of $X$ and let $|σ_1|:=\max_{1\le i\le n}|σ_i|$ be the spectral radius. Set $Y_n=\sqrt{4 n γ_n}\left[|σ_1|-1-\sqrt{\frac{γ_n}{4 n}}\right],$ where
$γ_{n}=\log{n}-2\log{\log{n}}-\log{2π}.$ As established in \cite{Cipolloni23Universality}, with specific moment-related conditions imposed on $x,$ the Gumbel distribution $Λ$ is identified as the universal weak limit of $Y_n.$ Subsequently, we extend this line of research and rigorously prove that the convergence rate, previously obtained for complex Ginibre ensembles in \cite{MaMeng25}, also possesses the property of universality. Precisely, one gets
$$\sup_{x\in \mathbb{R}}|\mathbb{P}(Y_n \leq x)-e^{-e^{-x}}|=\frac{2\log\log n}{e\log n}(1+o(1))$$
and
$$W_1\left(\mathcal{L}(Y_n), Λ\right)=\frac{2\log\log n}{\log n}(1+o(1))$$
for sufficiently large $n$, where $\mathcal{L}(Y_n)$ is the distribution of $Y_n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_03198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universality of the convergence rate for spectral radius of complex IID random matrices Hu, Xinchen Ma, Yutao Probability 60G70, 60B10, 15B52 Let $X$ be an $n\times n$ matrix with independent and identically distributed entries $x_{ij} \stackrel{\text { d }}{=} n^{-1 / 2} x$ for some complex random variable $x$ of mean zero and variance one. Let $\{σ_i\}_{1\le i\le n}$ be the eigenvalues of $X$ and let $|σ_1|:=\max_{1\le i\le n}|σ_i|$ be the spectral radius. Set $Y_n=\sqrt{4 n γ_n}\left[|σ_1|-1-\sqrt{\frac{γ_n}{4 n}}\right],$ where $γ_{n}=\log{n}-2\log{\log{n}}-\log{2π}.$ As established in \cite{Cipolloni23Universality}, with specific moment-related conditions imposed on $x,$ the Gumbel distribution $Λ$ is identified as the universal weak limit of $Y_n.$ Subsequently, we extend this line of research and rigorously prove that the convergence rate, previously obtained for complex Ginibre ensembles in \cite{MaMeng25}, also possesses the property of universality. Precisely, one gets $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Y_n \leq x)-e^{-e^{-x}}|=\frac{2\log\log n}{e\log n}(1+o(1))$$ and $$W_1\left(\mathcal{L}(Y_n), Λ\right)=\frac{2\log\log n}{\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Y_n)$ is the distribution of $Y_n$. |
| title | Universality of the convergence rate for spectral radius of complex IID random matrices |
| topic | Probability 60G70, 60B10, 15B52 |
| url | https://arxiv.org/abs/2505.03198 |