# distinguishing random orthogonal matrix from Gaussian random matrix

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? Have there been any results on this?

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Just because the distance goes to zero for $m=o(\sqrt{n})$ does not mean that it does NOT go to zero for bigger $m$ (for example, any $m=o(n)$). –  Igor Rivin Dec 2 '12 at 18:37
A nonzero total variation distance doesn't necessarily mean that the distributions can be told apart from samples. The place where the distributions differ substantially might take up a vanishingly small fraction of the probability weight of the distributions. –  Carl Feynman Dec 2 '12 at 21:05
Igor Rivin: The paper proved that it doesn't go to $0$ when m is a the order of $\sqrt n$. –  user14432 Dec 3 '12 at 0:58
Carl Feynman: Sorry, I should say that they could be told apart with probability at least a constant. –  user14432 Dec 3 '12 at 0:59
If that's true [the response to @Carl Feynman], don't you have an answer to your question? –  Igor Rivin Dec 3 '12 at 2:04