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Rank $1$ perturbations in random matrix theory -- a review of exact results | | Peter J. Forrester
; | | Date: |
2 Jan 2022 | | Abstract: | A number of random matrix ensembles permitting exact determination of their
eigenvalue and eigenvector statistics maintain this property under a rank $1$
perturbation. Considered in this review are the additive rank $1$ perturbation
of the Hermitian Gaussian ensembles, the multiplicative rank $1$ perturbation
of the Wishart ensembles, and rank $1$ perturbations of Hermitian and unitary
matrices giving rise to a two-dimensional support for the eigenvalues. The
focus throughout is on exact formulas, which are typically the result of
various integrable structures. The simplest is that of a determinantal point
process, with others relating to partial differential equations implied by a
formulation in terms of certain random tridiagonal matrices. Attention is also
given to eigenvector overlaps in the setting of a rank $1$ perturbation. | | Source: | arXiv, 2201.00324 | | Services: | Forum | Review | PDF | Favorites | |
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