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#### The statistics of spectral shifts due to finite rank perturbations

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##### Citation

Dietz, B., Schanz, H., Smilansky, U., & Weidenmüller, H. A. (2021). The statistics
of spectral shifts due to finite rank perturbations.* Journal of Physics A,* *54*(1):
015203. doi:10.1088/1751-8121/abc9da.

Cite as: https://hdl.handle.net/21.11116/0000-000A-3CBE-E

##### Abstract

This article is dedicated to the following class of problems. Start with

an N x N Hermitian matrix randomly picked from a matrix ensemble-the

reference matrix. Applying a rank-t perturbation to it, with t taking

the values 1 <= t <= N, we study the difference between the spectra of

the perturbed and the reference matrices as a function of t and its

dependence on the underlying universality class of the random matrix

ensemble. We consider both, the weaker kind of perturbation which either

permutes or randomizes t diagonal elements and a stronger perturbation

randomizing successively t rows and columns. In the first case we derive

universal expressions in the scaled parameter tau = t/N for the

expectation of the variance of the spectral shift functions, choosing as

random-matrix ensembles Dyson's three Gaussian ensembles. In the second

case we find an additional dependence on the matrix size N.

an N x N Hermitian matrix randomly picked from a matrix ensemble-the

reference matrix. Applying a rank-t perturbation to it, with t taking

the values 1 <= t <= N, we study the difference between the spectra of

the perturbed and the reference matrices as a function of t and its

dependence on the underlying universality class of the random matrix

ensemble. We consider both, the weaker kind of perturbation which either

permutes or randomizes t diagonal elements and a stronger perturbation

randomizing successively t rows and columns. In the first case we derive

universal expressions in the scaled parameter tau = t/N for the

expectation of the variance of the spectral shift functions, choosing as

random-matrix ensembles Dyson's three Gaussian ensembles. In the second

case we find an additional dependence on the matrix size N.