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

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 ($\min \{i, n-i\}=Θ(n)$), $$\E |λ_i(M_n)-\sqrt{n} γ_i|^2 = O(n^{-c}). $$ \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 $λ_i(M_n)$ 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.