[Paper Review] Almost-optimal bulk regularity conditions in the CLT for Wigner matrices

Benjamin Landon, Philippe Sosoe|arXiv (Cornell University)|Apr 7, 2022

Random Matrices and Applications4 citations

TL;DR

This paper establishes almost-optimal regularity conditions for the central limit theorem (CLT) in linear spectral statistics of Wigner matrices, showing that Gaussian fluctuations hold for test functions in Sobolev space $ H^{1/2 + \varepsilon} $ or Hölder space $ C^{1/2 + \varepsilon} $, even with low regularity, provided they are supported in the bulk of the semicircle law. The key result is that the limiting variance matches the classical CLT expression, confirming Johansson's conjecture near the critical regularity threshold $ s = 1/2 $.

ABSTRACT

We consider linear spectral statistics of the form $\mathrm{tr} ( φ(H))$ for test functions $φ$ of low regularity and Wigner matrices $H$ with smooth entry distribution. We show that for functions $φ$ in the Sobolev space $H^{1/2+\varepsilon}$ or the space $C^{1/2+\varepsilon}$, that are supported within the spectral bulk of the semicircle distribution, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\varepsilon >0$.

Motivation & Objective

• Address the conjecture that the central limit theorem (CLT) for linear spectral statistics of Wigner matrices holds under the weakest possible regularity conditions, specifically when the limiting variance is finite.
• Close the gap between known CLT results (e.g., $ H^{3/2+\varepsilon} $, $ C^1 $) and the conjectured optimal threshold $ H^{1/2} $ or $ C^{1/2} $.
• Establish that the classical variance formula in the CLT remains valid for functions with regularity as low as $ H^{1/2 + \varepsilon} $ or $ C^{1/2 + \varepsilon} $, for any $ \varepsilon > 0 $, within the spectral bulk.
• Provide a rigorous analysis of the covariance structure of linear spectral statistics under low regularity, using advanced tools from stochastic analysis and spectral theory.
• Confirm that the critical regularity threshold $ s = 1/2 $ is nearly optimal, with the variance functional remaining positive and well-defined for such functions.

Proposed method

• The analysis relies on a decomposition of the spectral statistics into sub-microscopic, microscopic, and mesoscopic frequency components, each treated with tailored estimates.
• Use of Littlewood-Paley theory to decompose test functions into frequency dyadic blocks, enabling control of the singular double integral term $ \int \int \frac{(\varphi(x) - \varphi(y))^2}{(x-y)^2} dx dy $, which characterizes the variance in the CLT.
• Application of homogenization techniques for Dyson Brownian motion to derive self-consistent equations for the covariance of linear statistics, tracking the evolution of fluctuations under stochastic dynamics.
• Employment of Stein’s method for normal approximation, with a self-consistent equation for the error term in the Stein’s method framework, expanded up to seventh order in the perturbation series.
• Derivation of precise asymptotic expansions for Hermite polynomials and related integrals using Stirling’s approximation and Laplace method, crucial for estimating the variance in the Gaussian ensemble case.
• Verification of the positivity of the variance functional via Chebyshev polynomial expansions and moment inequalities, ensuring the limiting variance is well-defined and non-degenerate.

Experimental results

Research questions

• RQ1 Can the central limit theorem for linear spectral statistics of Wigner matrices be extended to test functions with regularity as low as $ H^{1/2 + \varepsilon} $ or $ C^{1/2 + \varepsilon} $, for any $ \varepsilon > 0 $?
• RQ2 Does the classical variance formula for the CLT remain valid under such low regularity conditions, particularly in the bulk of the semicircle law?
• RQ3 Is the double integral term $ \int_{-2}^2 \int_{-2}^2 \frac{(\varphi(x) - \varphi(y))^2}{(x-y)^2} \frac{4 - xy}{\sqrt{4 - x^2}\sqrt{4 - y^2}} dx dy $ still the dominant contribution to the variance at this regularity threshold?
• RQ4 Can the variance functional remain positive and well-defined for functions in $ H^{1/2 + \varepsilon} $, ensuring the limiting Gaussian distribution is non-degenerate?
• RQ5 Does the fourth cumulant $ s_4 $ of the matrix entries affect the variance in the low-regularity regime, and is the classical correction term still valid?

Key findings

• The paper establishes the CLT for linear spectral statistics of Wigner matrices with test functions in the Sobolev space $ H^{1/2 + \varepsilon} $ for any $ \varepsilon > 0 $, confirming the conjecture that this is nearly optimal regularity.
• An identical CLT holds for functions in the Hölder space $ C^{1/2 + \varepsilon} $, demonstrating that the critical regularity threshold $ s = 1/2 $ is nearly sharp.
• The limiting variance of the linear spectral statistics matches the classical expression $ V_H(\varphi) $, even for functions of regularity $ H^{1/2 + \varepsilon} $, with the same double integral and cumulant correction terms.
• Using Littlewood-Paley theory, the authors show that the most singular term $ \int \int \frac{(\varphi(x) - \varphi(y))^2}{(x-y)^2} dx dy $ is finite and controls the variance for $ \varphi \in H^{1/2 + \varepsilon} $, linking this to the homogeneous Sobolev norm.
• The variance functional is proven to be positive for $ N \geq 4 $, ensuring the limiting Gaussian distribution is non-degenerate under the given regularity conditions.
• Through asymptotic analysis of Hermite polynomials and integrals, the authors derive precise estimates that validate the variance formula down to the $ H^{1/2 + \varepsilon} $ threshold, closing a long-standing gap in the theory.