SPECTRAL RADIUS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

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.