Jiang's paper (http://projecteuclid.org/euclid.aop/1158673325) shows the following: Suppose that $G$ is an $n\times n$ random matrix with entries i.i.d. $N(0,1/n)$, and $Z$ is a random $n\times n$ orthogonal matrix (uniform w.r.t. Haar measure). Let $D_1$ be the distribution of topleft $m \times m$ block of $G$, and $D_2$ be the distribution of topleft $m\times m$ block of $Z$. Then the total variance distance between $D_1$ and $D_2$ goes to 0 if $m=o(\sqrt n)$ and the t.v.d. is at least a constant if $m$ is at the order of $\sqrt n$.
This suggests that, given a random matrix which comes from the distribution of $G$ or the distribution of $Z$, we can tell which case it is with probability, say, $\geq 3/4$, by reading its top-left $\sqrt{n}\times\sqrt{n}$ block (dimension up to a constant factor).
My question is, What property should we examine on the top-left $\sqrt{n}\times\sqrt{n}$ block? Operator norm? Has Have there been any results on this?
