Let $X \in \mathbb{R}^{p\times p}$ be a large square matrix, consisting of i.i.d. Gaussian entries. Then it is known that the singular values of $X$ follow the Marchenko-Pastur law.


Now let's introduce an adversary, who arbitrarily selects $p/2$ rows of X and form a submatrix. Can the adversary succeed to make the singular values of this matrix look very different from the Marchenko-Pastur law of a $\frac{p}{2} \times p$ Gaussian matrix?



Restricted singular values of random matrix
It depends what do you mean by "look very different". I assume that you mean 
that the empirical measure is close to that of the MP law. The answer below 
assumes this is what you meant.


Short answer: no.


Longer answer: there is an exponential (in $p$) number of ways to choose the rows. But the
concentration of the empirical measure is exponential in $p^2$. 


