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May 2019 Location of the spectrum of Kronecker random matrices
Johannes Alt, László Erdős, Torben Krüger, Yuriy Nemish
Ann. Inst. H. Poincaré Probab. Statist. 55(2): 661-696 (May 2019). DOI: 10.1214/18-AIHP894


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.

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.


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Johannes Alt. László Erdős. Torben Krüger. Yuriy Nemish. "Location of the spectrum of Kronecker random matrices." Ann. Inst. H. Poincaré Probab. Statist. 55 (2) 661 - 696, May 2019.


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

zbMATH: 07097327
MathSciNet: MR3949949
Digital Object Identifier: 10.1214/18-AIHP894

Primary: 15B52, 60B20

Rights: Copyright © 2019 Institut Henri Poincaré


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Vol.55 • No. 2 • May 2019
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