Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles

Gernot Akemann, Tomasz Checinski, Dang-Zheng Liu, and Eugene Strahov

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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.


Les perturbations de rang fini des trois ensembles de matrices aléatoires complexes rectangulaires suivants sont comparés: d’abord un ensemble de Wishart généralisé, avec une matrice aléatoire et deux matrices de corrélation fixées, introduit par Borodin et Péché ; ensuite le produit de deux matrices aléatoires indépendantes, dont une a des éléments corrélés ; enfin le cas où deux matrices aléatoires sont couplées par une matrice fixée. Nous prouvons que la statistique des valeurs singulières des trois ensembles est déterminantale et nous dérivons des représentations en termes d’intégrales de contour doubles pour leurs noyaux respectifs. Dans la limite de dimension de matrice infinie à l’origine du spectre, on trouve trois noyaux différents, qui dépendent de la perturbation du rang fini des matrices de corrélation et du couplage et s’avèrent être intégrables. Le premier noyau (I) est trouvé pour le cas de deux matrices indépendantes du second ensemble, et pour celui de deux matrices faiblement couplées du troisième ensemble. Ce noyau généralise celui du type Meijer-G, valable pour deux matrices indépendantes et non corrélées. Le troisième noyau (III) est obtenu pour l’ensemble de Wishart généralisé et pour deux matrices couplées de façon forte. Celui-là généralise le noyau de Bessel perturbé de Desrosiers et Forrester. Finalement, le noyau (II), qui est trouvé pour l’ensemble de deux matrices couplées, représente une interpolation entre les noyaux (I) et (III), ce qui généralise des résultats précédemment obtenus par certains des auteurs.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 441-479.

Received: 5 May 2017
Revised: 19 October 2017
Accepted: 23 January 2018
First available in Project Euclid: 18 January 2019

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Zentralblatt MATH identifier

Primary: 15A52 60G55: Point processes 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Products of random matrices Correlated Wishart ensemble Determinantal point processes Biorthogonal ensembles Finite rank perturbations


Akemann, Gernot; Checinski, Tomasz; Liu, Dang-Zheng; Strahov, Eugene. Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 441--479. doi:10.1214/18-AIHP888.

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