Circular law for random matrices with unconditional log-concave distribution

Radosław Adamczak (1) , Djalil Chafai (2, 3)

1 MIMUW - Faculty of Mathematics, Informatics, and Mechanics [Warsaw]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Concentration of measure; Convex bodies; Random Matrices; Spectral Analysis

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Radosław Adamczak, Djalil Chafai. Circular law for random matrices with unconditional log-concave distribution. Communications in Contemporary Mathematics, 2015, 17 (4), pp.1550020. ⟨hal-00803841⟩