Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908519625654272 |
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| author | Hu, Xinchen Ma, Yutao |
| author_facet | Hu, Xinchen Ma, Yutao |
| contents | Let $X$ be a real $(β=1)$ or complex $(β=2)$ Ginibre ensemble. Let $\{σ_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\max_i \Re σ_i.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $Λ_β$ with distribution function $e^{-\fracβ{2}e^{-x}}.$ We further prove that
$$\sup_{x\in \mathbb{R}}|\mathbb{P}(Z_n \leq x)-e^{-\fracβ{2}e^{-x}}|=\frac{25\log \log n}{4e \log n}(1+o(1))$$
and
$$ W_1\left(\mathcal{L}(Z_n), Λ_β\right)=\frac{25\log \log n}{4\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Z_n)$ is the distribution of $Z_n$ and $W_1$ is the Wasserstein distance. Similar results hold for $\max_{i} |σ_i|.$ Furthermore, the convergence rates of the complex Ginibre ensemble are universal for complex iid random matrices under certain moment conditions on entries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution Hu, Xinchen Ma, Yutao Probability 60G70, 60B20, 60B10 Let $X$ be a real $(β=1)$ or complex $(β=2)$ Ginibre ensemble. Let $\{σ_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\max_i \Re σ_i.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $Λ_β$ with distribution function $e^{-\fracβ{2}e^{-x}}.$ We further prove that $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Z_n \leq x)-e^{-\fracβ{2}e^{-x}}|=\frac{25\log \log n}{4e \log n}(1+o(1))$$ and $$ W_1\left(\mathcal{L}(Z_n), Λ_β\right)=\frac{25\log \log n}{4\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Z_n)$ is the distribution of $Z_n$ and $W_1$ is the Wasserstein distance. Similar results hold for $\max_{i} |σ_i|.$ Furthermore, the convergence rates of the complex Ginibre ensemble are universal for complex iid random matrices under certain moment conditions on entries. |
| title | Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution |
| topic | Probability 60G70, 60B20, 60B10 |
| url | https://arxiv.org/abs/2506.04560 |