Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Location of the spectrum of Kronecker random matrices

Johannes Alt, László Erdős, Torben Krüger, and Yuriy Nemish

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Abstract

For a general class of large non-Hermitian random block matrices ${\boldsymbol X}$ we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of ${\boldsymbol X}$ as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.

Résumé

Pour une classe générale de grandes matrices aléatoires par blocs non hermitiennes ${\boldsymbol X}$, nous montrons qu’avec très grande probabilité, il n’y a pas de valeurs propres en dehors d’un ensemble déterministe. Cet ensemble est obtenu à partir de l’équation de Dyson pour l’hermitisation de ${\boldsymbol X}$ comme l’approximation auto-cohérente du pseudo-spectre. Nous démontrons que l’analyse de l’équation de Dyson provenant de (Probab. Theory Related Fields (2018)) permet d’étudier de façon unifiée de nombreux ensembles de matrices structurées.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 661-696.

Dates
Received: 15 July 2017
Revised: 25 January 2018
Accepted: 23 February 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820827

Digital Object Identifier
doi:10.1214/18-AIHP894

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

Keywords
Outliers Block matrices Local law Non-Hermitian random matrix Self-consistent pseudospectrum

Citation

Alt, Johannes; Erdős, László; Krüger, Torben; Nemish, Yuriy. Location of the spectrum of Kronecker random matrices. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 661--696. doi:10.1214/18-AIHP894. https://projecteuclid.org/euclid.aihp/1557820827


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