On the spectral radius of a random matrix: An upper bound without fourth moment
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
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01346261
Journal nameAnnals of Probability
MetadataShow full item record
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 / KeywordsCombinatorics; Digraph; Spectral Radius; Random matrix; Heavy Tail
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