The Annals of Probability

On the spectral radius of a random matrix: An upper bound without fourth moment

Charles Bordenave, Pietro Caputo, Djalil Chafaï, and Konstantin Tikhomirov

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Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance. In other words, that there are no outliers in the circular law. In this work, we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.

Article information

Ann. Probab., Volume 46, Number 4 (2018), 2268-2286.

Received: July 2016
Revised: May 2017
First available in Project Euclid: 13 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C20: Directed graphs (digraphs), tournaments 15B52: Random matrices 47A10: Spectrum, resolvent 05C80: Random graphs [See also 60B20]

Combinatorics digraph spectral radius random matrix heavy tail


Bordenave, Charles; Caputo, Pietro; Chafaï, Djalil; Tikhomirov, Konstantin. On the spectral radius of a random matrix: An upper bound without fourth moment. Ann. Probab. 46 (2018), no. 4, 2268--2286. doi:10.1214/17-AOP1228.

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