Optimal W₁ and Berry-Esseen Bounds for the Spectral Radius of Large Chiral Non-Hermitian Random Matrices

Abstract

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≤i≤n be its n eigenvalues with positive real part. Set

$X_{n}:={\sqrt{log\;s_{n}}}\left(\frac{\text{2}n\;\text{max}_{1\leq{i}\leq{n}}\mid\zeta_{i}\mid^{2}\;-\;2{\sqrt{n(n+v)}}}{\sqrt{2n+v}}-a(s_{n})\right)$

with $s_{n}=\frac{n(n+v)}{2n+v}$ and $a(s_{n})=\sqrt{log\;s_{n}}-\frac{\text{log}(\sqrt{2\pi}\;\text{log}\;s_{n})}{\sqrt{log\;s_{n}}}$. It was proved in [J. Math. Phys., 2020, 61(1): Paper No. 013508, 31 pp.] that X_n converges weakly to the Gumbel distribution Λ. In this paper, we further prove that

$\lim_{n \rightarrow \infty}\frac{\text{log}\;s_{n}}{(\text{log}\;\text{log}\;s_{n})^{2}}W_{1}\;({\cal{L}}(X_{n}),\Lambda)=\frac{1}{2}$

and the Berry-Esseen bound

$\lim_{n \rightarrow \infty}\frac{\text{log}\;s_{n}}{(\text{log}\;\text{log}\;s_{n})^{2}}{_{x\in{\mathbb{R}}}^{\text{sup}}}\mid{F}_{n}(x)-e^{-e^{-x}}\mid=\frac{1}{2e}.$

Here, ${\cal{L}}(X_{n})$ and F_n are the distribution and distribution function of X_n, respectively, and v = v_n is an arbitrary sequence of non-negative integers.

Keywords

Gumbel distribution / spectral radius / large chiral random matrix / Wasserstein distance / 60G70 / 60B20 / 60B10

References

[1] Abramowitz M, Stegun IA. Handbook of Mathematical Functions, 1972, New York: Dover Publications

[2] Akemann G. The complex Laguerre symplectic ensemble of non-Hermitian matrices. Nucl. Phys. B, 2005, 7303253299.

[3] J. Math. Phys., 2010, 5110

[4] Akemann G, Byun S-S, Kang N-G. A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality. Ann. Henri Poincaré, 2021, 22410351068.

[5] J. Phys. A, 2010, 438

[6] Anderson GW, Guionnet A, Zeitouni O. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, 118, 2010, Cambridge: Cambridge Univ. Press

[7] Bai ZD, Silverstein JW. Spectral Analysis of Large Dimensional Random Matrices, 2010Second EditionNew YorkSpringer

[8] Bordenave C, Caputo P, Chafaï D. Tikhomirov K., On the spectral radius of a random matrix: an upper bound without fourth moment. Ann. Probab., 2018, 46422682286.

[9] Bordenave C, Chafaï D, García-Zelada D. Convergence of the spectral radius of a random matrix through its characteristic polynomial. Probab. Theory Related Fields, 2022, 1823–411631181.

[10] Chafaï D. Around the circular law: an update, 2018

[11] Chafaï D, Péché S. A note on the second order universality at the edge of Coulomb gases on the plane. J. Stat. Phys., 2014, 1562368383.

[12] J. Math. Phys., 2020, 611

[13] Cipolloni G, Erdős L, Xu Y. Universality of extremal eigenvalues of large random matrices, 2023

[14] El Karoui N. A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab., 2006, 34620772117.

[15] Hu X, Ma Y. Convergence rate of extreme eigenvalue of Ginibre ensembles to the Gumbel distribution, 2025

[16] Johansson K. Shape fluctuations and random matrices. Comm. Math. Phys., 2000, 2092437476.

[17] Kostlan E. On the spectra of Gaussian matrices. Linear Algebra Appl., 1992, 162/164: 385-388.

[18] J. Stat. Mech. Theory Exp., 2018, 20181

[19] Random Matrices Theory Appl., 2026, 152

[20] Ma YT, Meng X. Quantitative analysis of rightmost eigenvalue for a large chiral non-Hermitian random matrix. Statistical Learning and Data Science, 20262

[21] Ma YT, Wang S. Deviation probabilities for extremal eigenvalues of large chiral non-Hermitian random matrices. Forum Math., 2025, 373717746.

[22] Marchenko VA, Pastur LA. Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.), 1967, 72114507536

[23] Olver FWJ, Lozier DW, Boisvert RF, Clark CW. NIST Handbook of Mathematical Functions, 2010, Cambridge: Cambridge Univ. Press

[24] Phys. Rev. Lett., 2004, 9322

[25] Petrov VV. Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, 1975, New York: Springer

[26] Rider BC. A limit theorem at the edge of a non-Hermitian random matrix ensemble. J. Phys. A, 2003, 361234013409.

[27] Rider BC, Sinclair CD. Extremal laws for the real Ginibre ensemble. Ann. Appl. Probab., 2014, 24416211651.

[28] Schnelli K., Xu Y., Quantitative Tracy-Widom laws for the largest eigenvalue of generalized Wigner matrices. Electron. J. Probab., 2023, 28: Paper No. 129, 38 pp.

[29] Villani C. Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften, 338, 2009, Berlin: Springer-Verlag

Rights & permissions

Peking University