# 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.