Universal Gaussian fluctuations of non-Hermitian matrix ensembles

Ivan Nourdin (1) , Giovanni Peccati (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 MODAL'X - Modélisation aléatoire de Paris X

We prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our techniques rely on a universality principle for the Gaussian Wiener chaos, recently proved by the authors together with Gesine Reinert, as well as on some combinatorial estimates. Unlike other related results in the probabilistic literature, we do not require that the law of the entries has a density with respect to the Lebesgue measure. In particular, our results apply to the ensemble of Bernoulli random matrices.

Keywords: Universality; Central limit theorems; Eigenvalues; Fourth-moment criteria; Invariance principles; Non-Hermitian random matrices; Normal approximation; Spectral moments; Universality.

Domains: Probability [math.PR]

Ivan Nourdin, Giovanni Peccati. Universal Gaussian fluctuations of non-Hermitian matrix ensembles. 2009. ⟨hal-00408877v3⟩