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 $n$.
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Yes, see Roman Vershynin's notes. 

