Optimal $W_1$ and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel (2501.08661v2)

Abstract: Consider the chiral non-Hermitian random matrix ensemble with parameters $n$ and $v$ and the non Hermiticity parameter $\tau=0$ and let $(\zeta_i){1\le i\le n}$ be its $n$ eigenvalues with positive $x$-coordinate. Set $$X_n:=\sqrt{\log s_n}\left(\frac{2n \max{1\le i\le n}|\zeta_i|^2-2\sqrt{n(n+v)}}{\sqrt{2n+v}}-a(s_{n})\right)$$ with $s_n=n(n+v)/(2n+v)$ and $a(s_n)=\sqrt{\log s_n}-\frac{\log(\sqrt{2\pi}\log s_n)}{\sqrt{\log s_n}}.$ It was proved in \cite{JQ} that $X_n$ converges weakly to the Gumbel distribution $\Lambda$. In this paper, we give in further that $$\lim_{n\to\infty} \frac{\log s_n}{(\log\log s_n)^2}W_1\left(F_n, \Lambda\right)=\frac{1}{2}$$ 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 $X_n.$

• The paper demonstrates that the optimal W1 distance converges to a constant of 1/2 when scaled with logarithmic terms.
• The study establishes a Berry–Esseen bound quantifying the convergence rate between the empirical spectral radius and the Gumbel distribution.
• The paper highlights the universality of the Gumbel distribution in describing extremal eigenvalue statistics in non-Hermitian random matrix ensembles.

Insights into Non-Hermitian Random Matrices and Gumbel Convergence

The paper "Optimal W1 and Berry-Esseen Bound Between the Spectral Radius of Large Chiral Non-Hermitian Random Matrices and Gumbel" by Yutao Ma and Siyu Wang explores the intricate relationships between the spectral properties of large chiral non-Hermitian random matrices and their convergence to the Gumbel distribution. The authors rigorously investigate the weak convergence of the spectral radius of these matrices and provide precise estimates for the Wasserstein-1 distance (W1) and the Berry-Esseen bound, advancing our understanding of the statistical properties of non-Hermitian matrices.

The paper focuses on random matrices that are non-Hermitian and exhibit complex spectral behavior due to their chiral structure. The paper considers the Dirac matrix, denoted here as D, constructed from parameterized Gaussian matrices. This matrix plays a significant role in quantum chromodynamics and presents a challenging yet insightful model for studying non-Hermitian random matrices. The spectral radius of D, defined by the maximum absolute value of its eigenvalues, is the primary object of interest and is found to converge in distribution to the Gumbel distribution under specific conditions.

Key Contributions and Results

1. Wasserstein-1 Distance: The authors demonstrate that the W1 distance between the empirical spectral radius distribution and the Gumbel distribution converges to a constant. Specifically, they establish that the limit of this distance, scaled appropriately by logarithmic terms, yields a value of 1/2. This result quantifies the rate of convergence and offers a metric for comparing empirical and limiting distributions in statistical mechanics and random matrix theory.
2. Berry-Esseen Bound: Complementing the W1 distance, the paper establishes a Berry-Esseen type bound for the deviation between the empirical distribution of the spectral radius and the Gumbel cumulative distribution function. The bound reveals a convergence rate involving a term proportional to (log log Sn)^2/log Sn, indicating a slow, yet systematic, alignment with the Gumbel distribution.
3. Spectral Radius Behavior and Universality: The authors reinforce the universality of the Gumbel distribution at the spectral radius, contrasting it with the well-studied Tracy-Widom distribution in Hermitian matrices' edge behaviors. The paper contributes to the ongoing exploration of universality classes and extremal eigenvalue distributions for general non-Hermitian ensembles.
4. Technical Innovations: Methodologically, the paper adapts and extends techniques from previous works on Hermitian and non-Hermitian matrices. Specific attention is given to the derivation of asymptotic estimates for eigenvalue distributions and the development of new probabilistic inequalities tailored to complex random matrices.

Implications and Future Directions

The implications of these findings are profound for the statistical physics and mathematics communities, where understanding the behavior of non-Hermitian systems is pivotal. By providing rigorous bounds on convergence metrics like W1 and Berry-Esseen, this work enables researchers to more precisely characterize the fluctuations of empirical distributions and test conjectures around universality in random matrix theory.

Future avenues of research may explore specific parameter regimes, such as varying the non-Hermiticity parameter or exploring more general classes of distributions for the matrix entries. Furthermore, these insights could be extended to analyze more complex systems where randomness and non-Hermitian dynamics play critical roles, such as in open quantum systems or in the paper of complex networks.

Overall, Ma and Wang's careful analysis offers robust contributions to random matrix theory's ongoing discourse, fortifying the bridge between mathematical theory and statistical applications.