Vol. 2, No. 2, 2021

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Spectral radius of random matrices with independent entries

Johannes Alt, László Erdős and Torben Krüger

Vol. 2 (2021), No. 2, 221–280
Abstract

We consider random n × n matrices X with independent and centered entries and a general variance profile. We show that the spectral radius of X converges with very high probability to the square root of the spectral radius of the variance matrix of X when n tends to infinity. We also establish the optimal rate of convergence; that is a new result even for general i.i.d. matrices beyond the explicitly solvable Gaussian cases. The main ingredient is the proof of the local inhomogeneous circular law (Ann. Appl. Probab. 28:1 (2018), 148–203) at the spectral edge.

Keywords
spectral radius, inhomogeneous circular law, cusp local law
Mathematical Subject Classification 2010
Primary: 15B52, 60B20
Milestones
Received: 6 February 2020
Revised: 15 October 2020
Accepted: 25 December 2020
Published: 22 May 2021
Authors
Johannes Alt
Section de mathématiques
Université de Genève
Geneve
Switzerland
László Erdős
Institute of Science and Technology Austria
Klosterneuburg
Austria
Torben Krüger
Department of Mathematical Sciences
University of Copenhagen
Copenhagen
Denmark