---
_id: '7423'
abstract:
- lang: eng
  text: 'We compare finite rank perturbations of the following three ensembles of
    complex rectangular random matrices: First, a generalised Wishart ensemble with
    one random and two fixed correlation matrices introduced by Borodin and Péché,
    second, the product of two independent random matrices where one has correlated
    entries, and third, the case when the two random matrices become also coupled
    through a fixed matrix. The singular value statistics of all three ensembles is
    shown to be determinantal and we derive double contour integral representations
    for their respective kernels. Three different kernels are found in the limit of
    infinite matrix dimension at the origin of the spectrum. They depend on finite
    rank perturbations of the correlation and coupling matrices and are shown to be
    integrable. The first kernel (I) is found for two independent matrices from the
    second, and two weakly coupled matrices from the third ensemble. It generalises
    the Meijer G-kernel for two independent and uncorrelated matrices. The third kernel
    (III) is obtained for the generalised Wishart ensemble and for two strongly coupled
    matrices. It further generalises the perturbed Bessel kernel of Desrosiers and
    Forrester. Finally, kernel (II), found for the ensemble of two coupled matrices,
    provides an interpolation between the kernels (I) and (III), generalising previous
    findings of part of the authors.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gernot
  full_name: Akemann, Gernot
  last_name: Akemann
- first_name: Tomasz
  full_name: Checinski, Tomasz
  last_name: Checinski
- first_name: Dangzheng
  full_name: Liu, Dangzheng
  id: 2F947E34-F248-11E8-B48F-1D18A9856A87
  last_name: Liu
- first_name: Eugene
  full_name: Strahov, Eugene
  last_name: Strahov
citation:
  ama: 'Akemann G, Checinski T, Liu D, Strahov E. Finite rank perturbations in products
    of coupled random matrices: From one correlated to two Wishart ensembles. <i>Annales
    de l’Institut Henri Poincaré, Probabilités et Statistiques</i>. 2019;55(1):441-479.
    doi:<a href="https://doi.org/10.1214/18-aihp888">10.1214/18-aihp888</a>'
  apa: 'Akemann, G., Checinski, T., Liu, D., &#38; Strahov, E. (2019). Finite rank
    perturbations in products of coupled random matrices: From one correlated to two
    Wishart ensembles. <i>Annales de l’Institut Henri Poincaré, Probabilités et Statistiques</i>.
    Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/18-aihp888">https://doi.org/10.1214/18-aihp888</a>'
  chicago: 'Akemann, Gernot, Tomasz Checinski, Dangzheng Liu, and Eugene Strahov.
    “Finite Rank Perturbations in Products of Coupled Random Matrices: From One Correlated
    to Two Wishart Ensembles.” <i>Annales de l’Institut Henri Poincaré, Probabilités
    et Statistiques</i>. Institute of Mathematical Statistics, 2019. <a href="https://doi.org/10.1214/18-aihp888">https://doi.org/10.1214/18-aihp888</a>.'
  ieee: 'G. Akemann, T. Checinski, D. Liu, and E. Strahov, “Finite rank perturbations
    in products of coupled random matrices: From one correlated to two Wishart ensembles,”
    <i>Annales de l’Institut Henri Poincaré, Probabilités et Statistiques</i>, vol.
    55, no. 1. Institute of Mathematical Statistics, pp. 441–479, 2019.'
  ista: 'Akemann G, Checinski T, Liu D, Strahov E. 2019. Finite rank perturbations
    in products of coupled random matrices: From one correlated to two Wishart ensembles.
    Annales de l’Institut Henri Poincaré, Probabilités et Statistiques. 55(1), 441–479.'
  mla: 'Akemann, Gernot, et al. “Finite Rank Perturbations in Products of Coupled
    Random Matrices: From One Correlated to Two Wishart Ensembles.” <i>Annales de
    l’Institut Henri Poincaré, Probabilités et Statistiques</i>, vol. 55, no. 1, Institute
    of Mathematical Statistics, 2019, pp. 441–79, doi:<a href="https://doi.org/10.1214/18-aihp888">10.1214/18-aihp888</a>.'
  short: G. Akemann, T. Checinski, D. Liu, E. Strahov, Annales de l’Institut Henri
    Poincaré, Probabilités et Statistiques 55 (2019) 441–479.
date_created: 2020-01-30T10:36:50Z
date_published: 2019-02-01T00:00:00Z
date_updated: 2023-09-06T14:58:39Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/18-aihp888
external_id:
  arxiv:
  - '1704.05224'
  isi:
  - '000456070200013'
intvolume: '        55'
isi: 1
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1704.05224
month: '02'
oa: 1
oa_version: Preprint
page: 441-479
publication: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
publication_identifier:
  issn:
  - 0246-0203
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
status: public
title: 'Finite rank perturbations in products of coupled random matrices: From one
  correlated to two Wishart ensembles'
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 55
year: '2019'
...