Universality in the bulk of the spectrum for complex sample covariance matrices

Péché, Sandrine

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 80-106.

On considère des matrices de covariance empirique complexes MN = (1/N)YY* où Y est une matrice de taille N × p dont les coefficients Yij, 1 ≤ i ≤ N, 1≤j ≤ p, sont des variables aléatoires i.i.d. de loi F. Sous certaines hypothèses de régularité et de décroissance sur F, on montre l'universalité de certaines statistiques locales de valeurs propres au milieu du spectre quand N → ∞ et limN→∞ p/N = γ pour tout réel γ ∈ (0, ∞).

We consider complex sample covariance matrices MN = (1/N)YY* where Y is a N × p random matrix with i.i.d. entries Yij, 1 ≤ i ≤ N, 1 ≤ j ≤ p, with distribution F. Under some regularity and decay assumptions on F, we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where N → ∞ and limN→∞ p/N = γ for any real number γ ∈ (0, ∞).

Classification : 60B20, 60B10, 60B12
Mots clés : random matrix, bulk universality, sample covariance matrices

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