Optimal $W_1$ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel
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| Format: | Preprint |
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2025
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| _version_ | 1866917893622464512 |
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| author | Ma, Yutao Wang, Siyu |
| author_facet | Ma, Yutao Wang, Siyu |
| contents | Consider the chiral non-Hermitian random matrix ensemble with parameters $n$ and $v$ and the non Hermiticity parameter $τ=0$ and let $(ζ_i)_{1\le i\le n}$ be its $n$ eigenvalues with positive $x$-coordinate. Set $$X_n:=\sqrt{\log s_n}\left(\frac{2n \max_{1\le i\le n}|ζ_i|^2-2\sqrt{n(n+v)}}{\sqrt{2n+v}}-a(s_{n})\right)$$ with $s_n=n(n+v)/(2n+v)$ and $a(s_n)=\sqrt{\log s_n}-\frac{\log(\sqrt{2π}\log s_n)}{\sqrt{\log s_n}}.$ It was proved in \cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $Λ$. In this paper, we give in further that $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}W_1\left(F_n, Λ\right)=\frac{1}{2}$$ and the Berry-Esseen bound $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}\sup_{x\in\mathbb{R}}|F_n(x)-e^{-e^{-x}}|=\frac{1}{2e}.$$
Here, $F_n$ is the distribution (function) of $X_n.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_08661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal $W_1$ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel Ma, Yutao Wang, Siyu Probability 60G70, 60B20, 60B10 Consider the chiral non-Hermitian random matrix ensemble with parameters $n$ and $v$ and the non Hermiticity parameter $τ=0$ and let $(ζ_i)_{1\le i\le n}$ be its $n$ eigenvalues with positive $x$-coordinate. Set $$X_n:=\sqrt{\log s_n}\left(\frac{2n \max_{1\le i\le n}|ζ_i|^2-2\sqrt{n(n+v)}}{\sqrt{2n+v}}-a(s_{n})\right)$$ with $s_n=n(n+v)/(2n+v)$ and $a(s_n)=\sqrt{\log s_n}-\frac{\log(\sqrt{2π}\log s_n)}{\sqrt{\log s_n}}.$ It was proved in \cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $Λ$. In this paper, we give in further that $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}W_1\left(F_n, Λ\right)=\frac{1}{2}$$ and the Berry-Esseen bound $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}\sup_{x\in\mathbb{R}}|F_n(x)-e^{-e^{-x}}|=\frac{1}{2e}.$$ Here, $F_n$ is the distribution (function) of $X_n.$ |
| title | Optimal $W_1$ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel |
| topic | Probability 60G70, 60B20, 60B10 |
| url | https://arxiv.org/abs/2501.08661 |