RANDOM MATRICES: SHARP CONCENTRATION OF EIGENVALUES

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Abstract
Let [ ] be a Wigner matrix whose entries have vanishing third moment, normalized so that the spectrum is concentrated in the interval [-2, 2]. We prove a concentration bound for NI = NI(Wn), the number of eigenvalues of Wn in an interval I. Our result shows that NI decays exponentially with standard deviation at most O( log O(1) n). This is best possible up to the constant exponent in the logarithmic term. As a corollary, the bulk eigenvalues are localized to an interval of width O( log O(1) n/n); again, this is optimal up to the exponent. These results strengthen recent results of Erdős, Yau and Yin (under the extra assumption of vanishing third moment). Our proof is relatively simple and relies on the Lindeberg replacement argument.

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