[Paper Review] Random matrices: Localization of the eigenvalues and the necessity of four moments

Terence Tao, Van Vu|arXiv (Cornell University)|May 17, 2010

Random Matrices and Applications15 references20 citations

TL;DR

This paper establishes precise bounds on the localization of eigenvalues for Wigner random matrices, showing that the expected squared deviation of eigenvalues from their classical locations scales as $ O(n^{-c}) $ in the bulk when the third moment vanishes. It further demonstrates that the four-moment condition in the Four Moment Theorem is necessary, as changes in the fourth moment shift the mean eigenvalue by $ \Theta(n^{-1/2}) $, and conjectures a precise asymptotic dependence on the fourth moment at the $ n^{-1/2} $ scale.

ABSTRACT

Consider the eigenvalues $λ_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law, one expects that $λ_i(M_n)$ concentrates around $γ_i \sqrt n$, where $\int_{-\infty}^{γ_i} ρ_{sc} (x) dx = \frac{i}{n}$ and $ρ_{sc}$ is the semicircular function. In this paper, we show that if the entries have vanishing third moment, then for all $1\le i \le n$ $$\E |λ_i(M_n)-\sqrt{n} γ_i|^2 = O(\min(n^{-c} \min(i,n+1-i)^{-2/3} n^{2/3}, n^{1/3+\eps})) ,$$ for some absolute constant $c>0$ and any absolute constant $\eps>0$. In particular, for the eigenvalues in the bulk ($\min \{i, n-i\}=Θ(n)$), $$\E |λ_i(M_n)-\sqrt{n} γ_i|^2 = O(n^{-c}). $$ oindent A similar result is achieved for the rate of convergence. As a corollary, we show that the four moment condition in the Four Moment Theorem is necessary, in the sense that if one allows the fourth moment to change (while keeping the first three moments fixed), then the \emph{mean} of $λ_i(M_n)$ changes by an amount comparable to $n^{-1/2}$ on the average. We make a precise conjecture about how the expectation of the eigenvalues vary with the fourth moment.

Motivation & Objective

• To establish sharp concentration bounds for eigenvalues of Wigner matrices under vanishing third moment.
• To quantify the dependence of eigenvalue expectations on the fourth moment of matrix entries.
• To demonstrate that the four-moment condition in the Four Moment Theorem is necessary, not just sufficient.
• To conjecture a precise asymptotic formula for the expected eigenvalue shift due to changes in the fourth moment.

Proposed method

• Uses Talagrand’s concentration inequality and refined moment estimates to bound the second moment of eigenvalue deviations from classical locations.
• Applies a combinatorial expansion of moments via 2-admissible paths and tree decompositions to analyze the contribution of higher-order cumulants.
• Derives a moment formula involving the function $ g(x) = \frac{1}{2\pi} \frac{x^4 - 4x^2 + 2}{\sqrt{4 - x^2}} $, which links eigenvalue shifts to the fourth moment via analytic continuation and Cauchy integral formulas.
• Introduces a normalized shift $ s_i = \sqrt{n}(\mathbb{E}\lambda_i - \mathbb{E}\lambda_i') - \frac{1}{4}(\gamma_i^3 - 2\gamma_i)\kappa_0 $ to isolate the fourth moment contribution.
• Employs Riemann integration and trapezoid rule approximations to relate discrete eigenvalue sums to integrals against the semicircular density $ \rho_{sc}(x) $.
• Uses the jump formula across the cut $[-2,2]$ to derive the moment generating function for the fourth moment correction.

Experimental results

Research questions

• RQ1 How does the fourth moment of matrix entries affect the expected location of eigenvalues in the bulk?
• RQ2 What is the precise scaling of the eigenvalue mean shift due to changes in the fourth moment?
• RQ3 Is the four-moment condition in the Four Moment Theorem necessary, or merely sufficient?
• RQ4 Can a sharp asymptotic expansion for $ \mathbb{E}\lambda_i $ be derived that includes the fourth moment at the $ n^{-1/2} $ scale?
• RQ5 How do higher-order cumulants and path decompositions in the moment expansion contribute to eigenvalue localization?

Key findings

• For Wigner matrices with vanishing third moment, the expected squared deviation of eigenvalues from their classical locations is $ O(n^{-c}) $ in the bulk, for some absolute $ c > 0 $.
• The rate of convergence of eigenvalues to their classical locations is $ O(n^{-c}) $ in expectation, improving upon previous $ O(n^{1/2 + \varepsilon}) $ bounds.
• If the fourth moment of the entries is allowed to vary while the first three moments are fixed, the mean eigenvalue shifts by $ \Theta(n^{-1/2}) $ on average.
• The paper conjectures that $ \mathbb{E}\lambda_i = \sqrt{n}\gamma_i + n^{-1/2}C_{i,n} + \frac{1}{4\sqrt{n}}(\gamma_i^3 - 2\gamma_i)\mathbb{E}\eta^4 + O_{\delta}(n^{-1/2 - c}) $, showing explicit dependence on the fourth moment.
• The analysis reveals that the fourth moment correction arises from a specific spectral function $ g(x) $, which is derived via analytic continuation and jump formulas in the complex plane.
• The results imply that the four-moment condition in the Four Moment Theorem is not only sufficient but necessary for universality of local eigenvalue statistics.