If we randomly pick $k\ll n$ columns from a fixed $n\times n$ matrix $A$, what can one say about the distribution of the 2-norm condition number of the resulting $n\times k$ matrices $A_k$?
I'd expect that knowing the singular values of $A$ would strongly restrict the distribution. (If all singular values of $A$ are identical, then this also holds for the $A_k$.)