Let $A$ be an n by n GOE (Gaussian Orthogonal Ensemble) matrix. Define $\|A\|_{op}=\sup_{\|v\|=1}\langle v, Av\rangle$.
We have the following concentration inequality: (see https://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwiz4d6LuuD6AhUHlIkEHWT_DhIQFnoECBIQAQ&url=https%3A%2F%2Fterrytao.files.wordpress.com%2F2011%2F02%2Fmatrix-book.pdf&usg=AOvVaw1wbeiabsuEb4DfHXHjoUJK)
there exist constants $C,c>0$ so that for $x>0$, $$ P(\|A\|_{op}-E[\|A\|_{op}]\ge x)\le Ce^{-cnx^2}. $$
We know that $E[\|A\|_{op}]\to 2$ a.s. (see https://projecteuclid.org/journals/annals-of-probability/volume-16/issue-4/Necessary-and-Sufficient-Conditions-for-Almost-Sure-Convergence-of-the/10.1214/aop/1176991594.full)
Question:
From above results, can we say that there exist constants $C,c>0$ so that for $x>0$ $$ P(\|A\|_{op}\ge x+2)\le Ce^{-cnx^2}. $$
So do we have $$ P(\|A\|_{op}\ge x)\le Ce^{-cn(x-2)^2}. $$