Spectral radius of random matrices with independent entries

Abstract

We consider random $n\times n$ 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 [arXiv:1612.07776] at the spectral edge.

45 pages; We corrected a few typos in the published version

References in corpus (4)

• Transition to chaos in random networks with cell-type-specific connectivity
• Edge Universality for non-Hermitian Random Matrices
• What drives transient behaviour in complex systems?
• Random matrices: Universality of ESDs and the circular law

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• Directional Extremal Statistics for Ginibre Eigenvalues
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• The least singular value of the general deformed Ginibre ensemble
• Convergence of the spectral radius of a random matrix through its characteristic polynomial
• Density of small singular values of the shifted real Ginibre ensemble
• Limiting eigenvalue distribution of the general deformed Ginibre ensemble