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

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

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

Type

Article accepté pour publication ou publié

External document link

https://hal.archives-ouvertes.fr/hal-01346261

Date

2018

Journal name

Annals of Probability

Volume

Number

Pages

2268-2286

Abstract (EN)

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.

Subjects / Keywords

Combinatorics; Digraph; Spectral Radius; Random matrix; Heavy Tail