I've been working through Terence Tao's text on random matrices, and there's a step in a proof that I am having trouble with. We want to show Proposition 2.3.19. The assumptions are $M$ symmetric with entries which are variance 1, mean 0, and $O(1)$ in magnitude. The eventual goal is to conclude that $E{||M||}_{op}$ is bounded below by $(2-o(1)) \sqrt{n}$.
Let's say we've shown that $E{||M||}^k_{op} \geq (C_{k/2} +o_{k}(1))n^{k/2}$. We want to conclude something about $E{||M||}_{op}$ from this, but the k inside of the expectation is bothersome. The proof says to combine the previous result with a corollary of Talagrand's theorem for the Gaussian decay of the operator norm in such a case; namely $Pr[\big|{||M||}_{op}-E{||M||}_{op}\big| > t] \leq a e^{-bt^2}$. The formula also holds for the median in place of the mean, if this is helpful.
Using the above two facts, we are supposed to deduce that $E{||M||}_{op} \geq (C_{k/2}^{1/k}+o_k(1))\sqrt{n}$
Conceptually for me, I am having trouble connecting concentration of measure about the mean with knowledge of the value of the mean, at least in the lower bound direction.
Much appreciated