I have a question about the non-self-adjoint case that was studied by Rajan and Abbott. Simulations suggest that the total number of outlier eigenvalues (at least in the “balanced” case where the components of \psi_n sum up to 0) scales like \sqrt(n) (up to n ~ 3000). But you say above that the outlier point process converges to the zeros of that random Laurent series which is independent of n, suggesting that the number of outliers is O(1), and not O(\sqrt(n)), and my simulations are not actually showing the very large n behavior.

Am I right (in concluding the number is O(1) based on what you say), or am I missing something?

I would really appreciate your answer.

Thanks.

Thanks!

]]>I discussed this identity at

https://terrytao.wordpress.com/2010/12/17/the-mesoscopic-structure-of-gue-eigenvalues/

]]>Thanks for the reference!

]]>Restricted rank modification of the symmetric eigenvalue problem: Theoretical considerations by Arbenz, Gander, Golub

the formula in your Remark 2.2 is referred to as the “modified Weinstein determinant” following Weinstein and Stenger’s “Methods for Intermediate Problems for Eigenvalues”.

Just FYI :-)

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