From Gaussian to Gumbel: extreme eigenvalues of complex Ginibre products with exact rates

2025/10/09 by Yutao Ma, Ma, Yutao, Meng, Xujia · 1 citation
Mathematics · #15B52 #60F10 #FOS: Mathematics #Mathematical functions and polynomials #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2510.07942

openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

Let Z₁, ⋯, Z_n denote the eigenvalues of the product ∏_j=1^k_n \boldsymbolA_j, where \\boldsymbolA_j\_{1 ≤ j ≤ kn} are independent n× n complex Ginibre matrices. Define α= lim_{n → ∞} (n)/(k_n). We prove that X_n, a suitably rescaled version of max_{1 ≤ j ≤ n} |Z_j|², converges weakly as follows: to a non-trivial distribution Φ_α for α∈ (0, +∞), to the Gumbel distribution when α= +∞, and to the standard normal distribution when α= 0. This result reveals a phase transition at the boundaries of α. Furthermore, we establish the exact rates of convergence in each regime.

Citations

• Revisit on the convergence rate of normal extremes
• How fast does spectral radius of truncated circular unitary ensemble converge?
• Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution
• Optimal W₁ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel
• Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution
• Phase transitions for infinite products of large non-Hermitian random matrices
• Gaussian fluctuations for products of random matrices
• Lyapunov exponent, universality and phase transition for products of random matrices
• Random matrix products: Universality and least singular values
• Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble
• Empirical Distribution of Scaled Eigenvalues for Product of Matrices from the Spherical Ensemble
• Real eigenvalue statistics for products of asymmetric real Gaussian matrices
• Products of Independent Gaussian Random Matrices
• Empirical Distributions of Eigenvalues of Product Ensembles
• Universality for products of random matrices I: Ginibre and truncated unitary cases
• Spectral Radii of Large Non-Hermitian Random Matrices
• Asymptotic Spectra of Matrix-Valued Functions of Independent Random Matrices and Free Probability
• On the Asymptotic Spectrum of Products of Independent Random Matrices
• From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

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• Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

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