On the lower bound of the spectral norm of symmetric random matrices with independent entries

Details On the lower bound of the spectral norm of symmetric random matrices with independent entries On the lower bound of the spectral norm of symmetric random matrices with independent entries

2007-06-06

arxiv.org/abs/0706.0748v2

Mathematics

Probability

Scientific paper

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \*\sigma - o(N^{-6/11+\epsilon}), $ where $\sigma^2 $ is the variance of the matrix entries and $\epsilon $ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\epsilon >0, $ one has $$ \|A_N\| =2 \*\sigma + o(N^{-6/11+\epsilon}) $$ with probability going to 1 as $N \to \infty. $

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