Random matrices: Localization of the eigenvalues and the necessity of four moments

TL;DR

This paper investigates the localization of eigenvalues of random Hermitian matrices with decaying entries, showing the necessity of the four moment condition for eigenvalue distribution stability, especially in the bulk.

Contribution

It establishes precise bounds on eigenvalue deviations under vanishing third moment conditions and demonstrates the necessity of the four moment condition in the Four Moment Theorem.

Findings

Eigenvalues concentrate around predicted locations with bounds depending on moments.

Vanishing third moment leads to specific eigenvalue localization rates.

Four moment condition is necessary for eigenvalue distribution stability.

Abstract

Consider the eigenvalues $\lambda_i(M_n)$ (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 $\lambda_i(M_n)$ concentrates around $\gamma_i \sqrt n$, where $\int_{-\infty}^{\gamma_i} \rho_{sc} (x) dx = \frac{i}{n}$ and $\rho_{sc}$ is the semicircular function. In this paper, we show that if the entries have vanishing third moment, then for all $1\le i \le n$ $$\E |\lambda_i(M_n)-\sqrt{n} \gamma_i|^2 = O(\min(n^{-c} \min(i,n+1-i)^{-2/3} n^{2/3}, n^{1/3+\eps})) ,$$ for some absolute constant $c>0$ and any absolute constant $\eps>0$. In particular, for the eigenvalues in the bulk ($\min \{i, n-i\}=\Theta (n)$), $$\E |\lambda_i(M_n)-\sqrt{n} \gamma_i|^2 = O(n^{-c}). $$ \noindent 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 $\lambda_i(M_n)$ 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.