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Convergence of the spectral radius of a random matrix through its characteristic polynomial

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Abstract

Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers.

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Authors and Affiliations

  1. Institut de Mathématiques de Marseille, CNRS, Aix-Marseille Université, Marseille, France

    Charles Bordenave & David García-Zelada

  2. CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, 75016, Paris, France

    Djalil Chafaï

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  1. Charles Bordenave
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  2. Djalil Chafaï
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  3. David García-Zelada
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Correspondence to Charles Bordenave.

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CB and DGZ are supported by the Grant ANR-16-CE40-0024.

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Bordenave, C., Chafaï, D. & García-Zelada, D. Convergence of the spectral radius of a random matrix through its characteristic polynomial. Probab. Theory Relat. Fields 182, 1163–1181 (2022). https://doi.org/10.1007/s00440-021-01079-9

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