How fast does spectral radius of truncated circular unitary ensemble converge?
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909654248849408 |
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| author | Ma, Yutao Meng, Xujia |
| author_facet | Ma, Yutao Meng, Xujia |
| contents | Let $z_1, \cdots, z_p$ be the eigenvalues of $A,$ which is the left-top $p\times p$ submatrix of an $n\times n$ Haar-invariant unitary matrix. Suppose there exist two constants $0<h_1<h_2<1$ such that $h_1<\frac pn<h_2.$ Then,
$$\sup_{x\in \mathbb{R}}|\mathbb{P}(X_n\le x)-e^{-e^{-x}}|=\frac{(\log \log n)^{2}}{2e\log n}(1+o(1))$$ and further
$$ W_{1}\left(\mathcal{L}(X_n),Λ\right)=\frac{(\log\log n)^2}{2\log n}(1+o(1))$$
for $n$ large enough. Here, $Λ$ is the Gumbel distribution and $\mathcal{L}(X_n)$ is the distribution of $X_n$ with $X_n$ being some rescaled version of $\max_{1\le i\le p}|z_i|,$ the spectral radius of $A.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | How fast does spectral radius of truncated circular unitary ensemble converge? Ma, Yutao Meng, Xujia Probability 60F10, 15B52 Let $z_1, \cdots, z_p$ be the eigenvalues of $A,$ which is the left-top $p\times p$ submatrix of an $n\times n$ Haar-invariant unitary matrix. Suppose there exist two constants $0<h_1<h_2<1$ such that $h_1<\frac pn<h_2.$ Then, $$\sup_{x\in \mathbb{R}}|\mathbb{P}(X_n\le x)-e^{-e^{-x}}|=\frac{(\log \log n)^{2}}{2e\log n}(1+o(1))$$ and further $$ W_{1}\left(\mathcal{L}(X_n),Λ\right)=\frac{(\log\log n)^2}{2\log n}(1+o(1))$$ for $n$ large enough. Here, $Λ$ is the Gumbel distribution and $\mathcal{L}(X_n)$ is the distribution of $X_n$ with $X_n$ being some rescaled version of $\max_{1\le i\le p}|z_i|,$ the spectral radius of $A.$ |
| title | How fast does spectral radius of truncated circular unitary ensemble converge? |
| topic | Probability 60F10, 15B52 |
| url | https://arxiv.org/abs/2506.16967 |