Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1.
Are there any results on the asymptotic behavior of its $i$-th largest singular value?
I am mostly interested when $i=\sqrt{n}$ or $i=n/\log(n)$, or in general when $i$ is some increasing function of $n$.



Scaling laws for singular values of random matrices
Yes, see [Roman Vershynin's notes.](http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf)


