Suppose I uniformly sample matrices X from the Gaussian Unitary Ensemble (GUE) with variance \sigma^2. Consider the Ky-Fan d norm, i.e. the sum of the singular values, of X. Let's call this Z=||X||\_1. What is the distribution of Z as a function of the dimension d and the variance \sigma^2? Really, all I want are estimates of the mean and a good tail bound, so maybe also the second moment.



Distribution of 1-norm for Gaussian Unitary Ensemble
Let's normalise the variance of the entries to be $1$. Then GUE asymptotically obeys the semicircular law, i.e., the eigenvalues (which equal the singular values, as GUE is Hermitian), after dividing by $\sqrt{n}$, are distributed according to the law $\frac{1}{2 \pi} (4 - x^2)^{1/2}\_+ dx$. So the Schatten $1$-norm (Ky Fan norm) should asymptotically equal $\sqrt{n}$ times n times the integral


\begin{equation\*}
\int \frac{|x|}{2 \pi} (4 - x^2)^{1/2}\_+ dx,
\end{equation\*}


which Wolfram alpha tells me is $8/3 \pi$. So the answer is $n^{3/2} ( 8 / 3 \pi + o(1) )$ with probability $1-o(1)$. Using an explicit convergence rate for GUE, one can probably replace the $o(1)$ with $O(n^{-c})$ for some explicit constant $c>0$.


Getting the variance may be within current technology - it's some integral of two-point correlations of GUE, which are known - but somewhat tedious. Higher moments should also (in principle) be computable. My guess is that the limiting distribution will be asymptotically gaussian, but I might be wrong about this (the central limit theorem doesn't apply directly because the eigenvalues are correlated with each other).


