THE LARGEST EIGENVALUES OF FINITE RANK DEFORMATION OF LARGE WIGNER MATRICES: CONVERGENCE AND NONUNIVERSALITY OF THE FLUCTUATIONS

Mireille Capitaine (1) , Catherine Donati-Martin (2) , Delphine Féral (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMV - Laboratoire de Mathématiques de Versailles
3 IMB - Institut de Mathématiques de Bordeaux

We investigate the asymptotic spectrum of complex or real Deformed Wigner matrices when the entries of the Hermitian (resp., symmetric) Wigner matrix have a symmetric law satisfying a Poincaré inequality. The perturbation is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of the perturbation are sufficiently far from zero, the corresponding eigenvalues of the deformed Wigner matrix almost surely exit the limiting semicircle compact sup- port as the size becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of the Wigner matrix. On the other hand, when the perturbation is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of the Wigner matrix.

Mots clés: Deformed Wigner matrices; asymptotic spectrum; Stieltjes transform; largest eigenvalues; fluctuations; nonuniversality

Domaines: Probabilités [math.PR]

Mireille Capitaine, Catherine Donati-Martin, Delphine Féral. THE LARGEST EIGENVALUES OF FINITE RANK DEFORMATION OF LARGE WIGNER MATRICES: CONVERGENCE AND NONUNIVERSALITY OF THE FLUCTUATIONS. Annals of Probability, 2009, 37 (1), pp.1-47. ⟨10.1214/08-AOP394⟩. ⟨hal-00939972⟩