Circular law for random matrices with unconditional log-concave distribution
Adamczak, Radosław; Chafaï, Djalil (2015), Circular law for random matrices with unconditional log-concave distribution, Communications in Contemporary Mathematics, 17, 4. 10.1142/S0219199715500200
TypeArticle accepté pour publication ou publié
Journal nameCommunications in Contemporary Mathematics
MetadataShow full item record
Laboratoire d'Analyse et de Mathématiques Appliquées [LAMA]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.
Subjects / KeywordsConcentration of measure; Convex bodies; Random Matrices; Spectral Analysis
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