RANDOM MATRICES: SHARP CONCENTRATION OF EIGENVALUES

Abstract

Let [Formula: see text] 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 N I = N I (W n ), the number of eigenvalues of W n in an interval I. Our result shows that N I 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.

Locations

• Random Matrices Theory and Application - View
• arXiv (Cornell University) - View - PDF
• Random Matrices Theory and Application - View
• arXiv (Cornell University) - View - PDF
• Random Matrices Theory and Application - View
• arXiv (Cornell University) - View - PDF