Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Outliers in the spectrum for products of independent random matrices

Natalie Coston, Sean O’Rourke, and Philip Matchett Wood

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Abstract

For fixed $m\geq 1$, we consider the product of $m$ independent $n\times n$ random matrices with iid entries as $n\to \infty $. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the $m$th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Tao (Probab. Theory Related Fields 155 (2013) 231–263) for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations.

Résumé

Pour un $m\geq 1$ fixé, nous considérons le produit de $m$ matrices aléatoires indépendantes de taille $n\times n$, à coefficients i.i.d., lorsque $n\to \infty $. Sous certaines hypothèses sur les coefficients de chaque matrice, il est connu que la loi empirique limite des valeurs propres est décrite par la puissance $m$-ième de la loi circulaire. De plus, cette même loi limite apparaît toujours si chacune des matrices i.i.d. est perturbée additivement par une erreur déterministe de rang borné. Néanmoins, les perturbations de rang borné peuvent créer quelques valeurs propres atypiques (outliers). Nous décrivons la localisation asymptotique de ces valeurs propres atypiques, ce qui généralise un résultat de Tao (Probab. Theory Related Fields 155 (2013) 231–263) dans la cas d’une seule matrice i.i.d. Nos méthodes nous permettent également de considérer d’autres types de perturbations, dont des perturbations multiplicatives.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1284-1320.

Dates
Received: 19 January 2018
Revised: 30 January 2019
Accepted: 7 May 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345638

Digital Object Identifier
doi:10.1214/19-AIHP1002

Mathematical Reviews number (MathSciNet)
MR4076784

Zentralblatt MATH identifier
07199898

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
Product random matrices Outlier eigenvalues Perturbed products Isotropic limit law

Citation

Coston, Natalie; O’Rourke, Sean; Wood, Philip Matchett. Outliers in the spectrum for products of independent random matrices. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1284--1320. doi:10.1214/19-AIHP1002. https://projecteuclid.org/euclid.aihp/1584345638


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