Mireille Capitaine

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

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 Poincare 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.

Free Bessel Laws

We introduce and study a remarkable family of real probability measures that we call free Bessel laws. These are related to the free Poisson law. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

Central limit theorems for eigenvalues of deformations of Wigner matrices

In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and sufficient condition on the associated normalized eigenvector so that the fluctuations of the corresponding eigenvalue of the deformed model are universal.

Outliers in the spectrum of large deformed unitarily invariant models

In this paper we characterize the possible outliers in the spectrum of large deformed unitarily invariant additive and multiplicative models, as well as the eigenvectors corresponding to them. We allow both the non-deformed unitarily invariant model and the perturbation matrix to have non-trivial limiting spectral measures and spiked outliers in their spectrum. We uncover a remarkable new phenomenon: a single spike can generate asymptotically several outliers in the spectrum of the deformed model. The free subordination functions play a key role in this analysis.

Onsager-Machlup functional for some smooth norms on Wiener space

SummaryIn this paper, we determine Onsager-Machlup functionals for a variety of norms on Wiener space which includes among others Hölder norms for every 0<α<1/2, as well as Besov or Sobolev type norms. We basically require the knowledge of the small ball probabilities for the Wiener measure and use versions of the norms which are rotationaly invariant on the range of the Brownian paths, a property of crucial importance in our approach.

Geometric interpretation of the cumulants for random matrices previously defined as convolutions on the symmetric group

We show that, dealing with an appropriate basis, the cumulants for N×N random matrices (A1,…, An), previously defined in [2] and [3], are the coordinates of \( \mathbb{E}\left\{ {\prod \left( {{A_1} \otimes \cdots \otimes {A_n}} \right)} \right\}, \)where II denotes the orthogonal projection of A1⊗…⊗An on the space of invariant vectors of M N ⊗n under the natural action of the unitary, respectively orthogonal, group. In this way we make the connection between [5] and [2], [3]. We also give a new proof in that context of the properties satisfied by these matricial cumulants.

Limiting Eigenvectors of Outliers for Spiked Information-Plus-Noise Type Matrices

We consider an Information-Plus-Noise type matrix where the Information matrix is a spiked matrix. When some eigenvalues of the random matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the spikes. Note that, in an Appendix, we present alternative versions of the earlier results of Bai and Silverstein (Random Matrices Theory Appl 1(1):1150004, 44, 2012) (“noeigenvalue outside the support of the deterministic equivalent measure”) and Capitaine (Indiana Univ Math J 63(6):1875–1910, 2014) (“exact separation phenomenon”) where we remove some technical assumptions that were difficult to handle.