[Paper Review] Quantitative CLT for linear eigenvalue statistics of Wigner matrices

Zhigang Bao, Yukun He|arXiv (Cornell University)|Mar 9, 2021

Random Matrices and Applications46 references4 citations

TL;DR

This paper establishes a near-optimal convergence rate for the central limit theorem (CLT) of linear eigenvalue statistics (LES) of $N\times N$ Wigner matrices in Kolmogorov-Smirnov distance. Using a probabilistic approach, it identifies two distinct convergence rates—$N^{-1/2+\varepsilon}$ and $N^{-1+\varepsilon}$—depending on the first Chebyshev coefficient of the test function and the third moment of diagonal entries, with a necessary and sufficient condition distinguishing them. By removing a non-universal contribution, the shifted LES achieve a unified $N^{-1+\varepsilon}$ rate across all test functions.

ABSTRACT

In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions $f\in C^5(\mathbb R)$, we show that the convergence rate is either $N^{-1/2+\varepsilon}$ or $N^{-1+\varepsilon}$, depending on the first Chebyshev coefficient of $f$ and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, we identify an explicit, non-universal contribution in the linear eigenvalue statistics, which is responsible for the slow rate $N^{-1/2+\varepsilon}$ for non-Gaussian ensembles. By removing this non-universal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate $N^{-1+\varepsilon}$ for all test functions.

Motivation & Objective

• To provide a quantitative description of the convergence rate for the CLT of linear eigenvalue statistics (LES) of Wigner matrices, which had previously only been established in non-quantitative asymptotic forms.
• To resolve the long-standing gap in understanding the speed of weak convergence in random matrix theory, particularly for non-Gaussian Wigner ensembles.
• To identify the precise conditions under which different convergence rates emerge, distinguishing between $N^{-1/2+\varepsilon}$ and $N^{-1+\varepsilon}$ rates.
• To isolate and remove a non-universal contribution in LES that causes the slower $N^{-1/2+\varepsilon}$ rate, thereby unifying the convergence rate across all test functions.

Proposed method

• A probabilistic approach is employed to analyze the fluctuation of linear eigenvalue statistics, avoiding reliance on explicit joint eigenvalue density formulas used in prior analytical methods.
• The authors derive a precise characterization of the convergence rate in Kolmogorov-Smirnov distance, leveraging moment estimates and martingale-type decompositions.
• Key technical tools include fluctuation averaging of Green's functions and resolvent identities to control error terms in the expansion of eigenvalue statistics.
• The analysis distinguishes two regimes based on the first Chebyshev coefficient $c_1^f$ of the test function and the third moment $a_3 = \mathbb{E}[h_d^3]$ of the diagonal entries.
• A non-universal correction term is explicitly identified and removed from the LES, leading to a unified convergence rate.
• The proof uses a decomposition of the characteristic function of the normalized LES and applies Taylor expansion and moment bounds to control error terms.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices in Kolmogorov-Smirnov distance?
• RQ2 Under what conditions does the convergence rate degrade to $N^{-1/2+\varepsilon}$, and when is it improved to $N^{-1+\varepsilon}$?
• RQ3 Is there a non-universal contribution in the linear eigenvalue statistics that causes the slower convergence rate for non-Gaussian Wigner ensembles?
• RQ4 Can the convergence rate be unified across all test functions by removing this non-universal component?
• RQ5 What are the matching lower bounds for the convergence rates in the general class of test functions considered?

Key findings

• The convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices is $N^{-1/2+\varepsilon}$ when the first Chebyshev coefficient $c_1^f$ and the third moment $a_3$ of the diagonal entries satisfy a specific non-degeneracy condition.
• When this condition fails, the convergence rate improves to $N^{-1+\varepsilon}$, and this condition is both necessary and sufficient for the faster rate.
• For a general class of $C^5$ test functions, matching lower bounds are established, confirming the near-optimality of the derived rates.
• An explicit, non-universal contribution in the linear eigenvalue statistics is identified as the source of the $N^{-1/2+\varepsilon}$ rate in non-Gaussian ensembles.
• After removing this non-universal term, the shifted linear eigenvalue statistics achieve a unified convergence rate of $N^{-1+\varepsilon}$ for all test functions.
• The results are established in the stronger Kolmogorov-Smirnov distance, improving upon prior works that used weaker distances like quadratic Kantorovich or moment-based convergence.