Timeline for Minimal perturbation of a Wigner matrix needed to produce an orthogonal top eigenvector

8 events

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Sep 13, 2020 at 21:04	history	edited	Carlo Beenakker	CC BY-SA 4.0	more informative title
Sep 13, 2020 at 21:00	history	edited	YCor		edited tags
Sep 13, 2020 at 19:57	comment	added	Daniel Li		Oh oh ok. But spacing between first and second eigenvalue is $N^{-1/6}$, right? This means before reaching that level, changing k entries contributes roughly $\sqrt{k}/N$. This is where we get $N^{5/3}$?
Sep 13, 2020 at 19:53	comment	added	Carlo Beenakker		the shift of an eigenvalue by the eigenvalue spacing creates independent eigenvectors, which would then become orthogonal for the reason mentioned in the question.
Sep 13, 2020 at 19:48	comment	added	Daniel Li		Thank you but how shifting largest eigenvalue by $\delta_N$ gives orthogonality? Ps. this is not supposed to be trivial because we are asked to genuinely think about it.
Sep 13, 2020 at 19:34	comment	added	Carlo Beenakker		this seems quite nontrivial to me; the top eigenvector refers to the largest eigenvalue, which has a Tracy-Widom distribution with a level spacing $\delta_N\propto N^{-2/3}$ -- larger than in the bulk of the spectrum, where the spacing scales $\propto N^{-1}$. The question would seem to ask for the minimal rank $k$ of a perturbation that shifts the largest eigenvalue by $\delta_N$, to obtain an orthogonal eigenvector.
Sep 13, 2020 at 18:51	comment	added	Daniel Li		Any directional or strategical suggestion is welcome.
Sep 13, 2020 at 18:21	history	asked	Daniel Li	CC BY-SA 4.0