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

Bordenave, Charles; Chafaï, Djalil; García-Zelada, David (2021), Convergence of the spectral radius of a random matrix through its characteristic polynomial, Probability Theory and Related Fields, p. 1-12. 10.1007/s00440-021-01079-9

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Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-03053734
Date
2021
Journal name
Probability Theory and Related Fields
Publisher
Springer
Pages
1-12
Publication identifier
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Author(s)
Bordenave, Charles
Institut de Mathématiques de Marseille [I2M]
Chafaï, Djalil cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
García-Zelada, David
Institut de Mathématiques de Marseille [I2M]
Abstract (EN)
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.
Subjects / Keywords
Random matrix; Spectral radius; Gaussian analytic function; Central limit theorem; Combinatorics; Digraph; Circular law