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arXiv · 2201.00324

Rank $1$ perturbations in random matrix theory -- a review of exact results

Peter J. Forrester
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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.

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Peter J. Forrester. 2022-01-02. Rank $1$ perturbations in random matrix theory -- a review of exact results. https://arxiv.org/abs/2201.00324

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