pr.probability - Distribution of 1-norm for Gaussian Unitary Ensemble

I'm not at all expert on random matrix stuff, but until someone more qualified pops up, would you be interested in some crude estimates on Z in the $n to infty$ asymptotic? Or do you really want something sharper in each dimension?

EDIT/UPDATE: OK, here's my hand-wavy argument,
it's been a while since I did any probability theory, so caveat lector and all that.

I'm going to use the GOE just because that's the one I know better and to save me worrying about stray scaling factors.

The idea is that for any $n times n$ matrices $S$ and $T$ we always have

$Vert SVert_1 Vert T Vert_{rm op} geq |{rm tr}(ST)|$

where the subscript 1 denotes Ky-Fan/Schatten 1-norm and the subscript "op" denotes usual operator norm. In particular, if $S=T$ is self-adjoint then

$Vert SVert_1 geq || S ||_2^2\, /\, || S ||_{rm op}$

Now when S is GOE(n,$sigma^2$) then $n^{-2} Vert SVert_2^2$ is strongly concentrated round its mean (which is $sigma^2$) -- it's the average of a bunch of independent random variables so we could use variance estimates and Chebyshev, or probably some stronger exponential tail estimates.

Also, when S is GOE(n,$sigma^2$) then $n^{-1/2} Vert SVert_{rm op}$ is strongly concentrated round $2sigma$ - one can get exponential tail estimates, at least for an upper bound of $(2+epsilon)sigma$ for any positive $epsilon$. I think this is folklore or a special case of Big Machinery, but as I said I have a more elementary proof, albeit one which is probably not original.

So, there is going to be a high probability (for $n$ large) that || S ||₂² is bigger than $(1-epsilon)sigma^2n^2$, and there is going to be a high probability (for $n$ large) that $Vert SVert_{rm op}$ is less than $(2+epsilon)sigma n^{1/2}$. On the intersection of these two events you're going to find that

$Vert SVert_1 geq (1-epsilon)n^2sigma^2 / (2+epsilon)sigma n^{1/2}$

which gives the lower bound I was claiming.