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Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution
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Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution
abstract
Consider the complex Ginibre ensemble, whose eigenvalues are $(\lambda_i)_{1\le i\le n}$ and the spectral radius $R_n=\max_{1\le i\le n}|\lambda_i|.$ Set $X_n=\sqrt{4 \gamma_{n}}(R_{n}-\sqrt{n}-\frac12\sqrt{\gamma_{n}})$ and $F_n$ be its distribution function, where $\gamma_{n}=\log n-2\log(\sqrt{2\pi}\log n).$ It was proved in \cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $\Lambda.$ We prove in further in this paper that $$\lim_{n\to\infty} \frac{\log n}{\log\log n}\, W_1\left(F_n, \Lambda\right)=2$$ and the Berry-Esseen bound $$\lim\limits_{n\to \infty} \frac{\log n}{\log\log n}\sup_{x\in \mathbb{R}}|F_{n}(x)-e^{-e^{-x}}|=\frac{2}{e}.$$
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Precise convergence rate of spectral radius of product of complex Ginibre
The suitably rescaled max |Z_j|^2 of eigenvalues of the product of k_n complex Ginibre matrices converges weakly to Phi_alpha for finite positive alpha = lim n/k_n, to Gumbel for alpha infinite, and to normal for alph...
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