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July 2018 On the spectral radius of a random matrix: An upper bound without fourth moment
Charles Bordenave, Pietro Caputo, Djalil Chafaï, Konstantin Tikhomirov
Ann. Probab. 46(4): 2268-2286 (July 2018). DOI: 10.1214/17-AOP1228


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.


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Charles Bordenave. Pietro Caputo. Djalil Chafaï. Konstantin Tikhomirov. "On the spectral radius of a random matrix: An upper bound without fourth moment." Ann. Probab. 46 (4) 2268 - 2286, July 2018.


Received: 1 July 2016; Revised: 1 May 2017; Published: July 2018
First available in Project Euclid: 13 June 2018

zbMATH: 1393.05130
MathSciNet: MR3813992
Digital Object Identifier: 10.1214/17-AOP1228

Primary: 05C20‎ , 05C80 , 15B52 , 47A10

Keywords: combinatorics , ‎digraph‎ , heavy tail , Random matrix , spectral radius

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 4 • July 2018
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