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Spectral radius of random matrices with independent entries

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arxiv 1907.13631 v5 pith:H7SZCC7N submitted 2019-07-31 math.PR math-phmath.FAmath.MP

classification math.PRmath-phmath.FAmath.MP
keywords spectralmatricesradiusentriesgeneralindependentrandomvariance
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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.

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Cited by 1 Pith paper

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  1. Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble

    math.PR 2019-08 accept novelty 6.0 of 10

    For shifted real or complex Ginibre matrices near the spectral edge, the least singular value has optimal tail probability of order x in the complex case and square-root x with a Gaussian imaginary-part damping in the...

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