Concentration of Eigenvalues of random matrices

by cbro   Last Updated January 14, 2018 08:19 AM

In this paper --> on Page 5 "3. Concluding Remarks". I am trying to make sense of the second point, which starts as "Our estimation from Theorem 1 is sharp...". I don't quite get how he arrived at "the probability that $\lambda_1$ (largest eigenvalue) exceeds its median by $t$ is at least $\Omega(e^{-O(t^2)})$.

This is what I understood so far. He gives the upper bound on $E(\lambda_1)$ and the lower bound on $\lambda_1$ to get the smallest deviation from the mean. But then $t$ is just a deviation parameter... how does he compute the probability that the least difference exceeds $t$? And how is he getting this $e^-$ something term? Has this got something to do with the relation of the binomial distribution with the normal distribution? Is he measuring $t$ somehow in terms of how many standard deviations away from the mean it is?

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