Random matrices: Sharp concentration of eigenvalues

By Terence Tao, Van H. Vu. 2012

49 citations recorded by the source corpus.

Let $W_n= \frac{1}{\sqrt n} M_n$ 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 Erdos, Yau and Yin (under the extra assumption of vanishing third

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/922c71a395a6c131181c9bfebc3490814ae2720f

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

The local semicircle law for a general class of random matrices (2012, background): “ntration bounds in [20].) This result was then extended to the spectral edges in [19]. (Some improvements over the estimates from [13] at the edges, for a special class of ensembles, were obtained in [30].) In [19], the identical distribution of the entries of H was not required, but the upper bound”
Random matrix theory in statistics: A review (2014, methodology): “The works of Tao and Vu (2011, 2012a) show that the rate of convergence is at most Oðn 1 log nÞ.”