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

TL;DR

This review paper discusses exact results in random matrix theory involving rank 1 perturbations across various ensembles, highlighting integrable structures and eigenvector overlaps.

Contribution

It provides a comprehensive overview of exact formulas and integrable structures for rank 1 perturbations in different random matrix ensembles.

Findings

Explicit formulas for eigenvalue distributions under rank 1 perturbations

Identification of integrable structures like determinantal point processes

Analysis of eigenvector overlaps in perturbed matrices

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.