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

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.

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