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This paper calculates the fluctuations of eigenvalues of polynomials on largeHaar unitaries cut by finite rank deterministic matrices. When the eigenvaluesare all simple, we can give a complete algorithm for computing thefluctuations. When multiple eigenvalues are involved, we present severalexamples suggesting that a general algorithm would be much more complex.

Fluctuations of eigenvalues of a polynomial on Haar unitary and finite rank matrices