random matrices – A “simple” homework question on Wigner matrix


The instructor proposed a the following statement in the passing and suggested that we think about it (although it is not required):

For any $N times N$ Wigner matrix, we replace $k$ entries with resampled copy (i.e. the new entries have exact same distribution as the original but are resampled independently). Suppose that $k/N^{5/3}toinfty$. Denote top eigenvector of original and resampled matrix as $v,v’$. Show that
$mathbb{E}|langle v,v’ rangle| to_N0$.

Attempt: the intuition is of course that because of resampling too many entries, the two eigenvectors become less and less correlated. And in high dimensions, two independent vectors distributed uniformly on a ball tend to be orthogonal. But I struggle to explain the $N^{5/3}$ threshold or to formulate a rigorous proof.