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October 3, 2025Open Access

Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

Key Points

Convergence of rescaled maximum eigenvalues to Gumbel distribution is established for Ginibre ensembles, and their rates are characterized.
It is shown that the Wasserstein distance between the distributions diminishes at a specific rate for large n, indicating strong convergence properties.
Universality of convergence rates is noted for complex Ginibre ensembles under certain conditions, broadening the implications of the findings.
The precision of the convergence rate is quantitatively described in terms of logarithmic factors and relates to behavior of complex iid random matrices.

Abstract

Let X be a real (β=1) or complex (β=2) Ginibre ensemble. Let \σᵢ\₁ ₈ ₍ be the eigenvalues of X, and Zₙ be some rescaled version of ᵢ σᵢ. It was proved that Zₙ converges weakly to the Gumbel distribution Λ_β with distribution function e^-β{2e^-x}. We further prove that ₗ ₑ|P (Zₙ x) -e^-β{2e^-x}|=25 n4e n (1+o (1) ) and W₁ (L (Zₙ), Λ_β) =25 n4 n (1+o (1) ) for sufficiently large n, where L (Zₙ) is the distribution of Zₙ and W₁ is the Wasserstein distance. Similar results hold for ₈ |σᵢ|. Furthermore, the convergence rates of the complex Ginibre ensemble are universal for complex iid random matrices under certain moment conditions on entries.

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Hu et al. (Thu,) studied this question.

synapsesocial.com/papers/68e02f40f0e39f13e7fa2883 https://doi.org/https://doi.org/10.48550/arxiv.2506.04560

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