45 research outputs found
Additive/multiplicative free subordination property and limiting eigenvectors of spiked additive deformations of Wigner matrices and spiked sample covariance matrices
When some eigenvalues of a spiked multiplicative resp. additive deformation
model of a Hermitian Wigner matrix resp. a sample covariance matrix separate
from the bulk, we study how the corresponding eigenvectors project onto those
of the perturbation. We point out that the inverse of the subordination
function relative to the free additive resp. multiplicative convolution plays
an important part in the asymptotic behavior
Strong asymptotic freeness for Wigner and Wishart matrices
We prove that any non commutative polynomial of r independent copies of
Wigner matrices converges a.s. towards the polynomial of r free semicircular
variables in operator norm. This result extends a previous work of Haagerup and
Thorbjornsen where GUE matrices are considered, as well as the classical
asymptotic freeness for Wigner matrices (i.e. convergence of the moments)
proved by Dykema. We also study the Wishart case
The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
In this paper, we investigate the asymptotic spectrum of complex or real
Deformed Wigner matrices defined by where
is an Hermitian (resp., symmetric) Wigner matrix whose
entries have a symmetric law satisfying a Poincar\'{e} inequality. The matrix
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 are sufficiently far from zero,
the corresponding eigenvalues of almost surely exit the limiting
semicircle compact support as the size becomes large. The corresponding
limits are universal in the sense that they only involve the variance of the
entries of . On the other hand, when 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 .Comment: Published in at http://dx.doi.org/10.1214/08-AOP394 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices
We investigate the asymptotic behavior of the eigenvalues of spiked
perturbations of Wigner matrices when the dimension goes to infinity. The
entries of the Hermitian Wigner matrix have a distribution which is symmetric
and satisfies a Poincar\'e inequality. The perturbation matrix is a
deterministic Hermitian matrix whose spectral measure converges to some
probability measure with compact support. We assume that this perturbation
matrix has a fixed number of fixed eigenvalues (spikes) outside the support of
its limiting spectral measure whereas the distance between the other
eigenvalues and this support uniformly goes to zero as the dimension goes to
infinity. We establish that only a particular subset of the spikes will
generate some eigenvalues of the deformed model which will converge to some
limiting points outside the support of the limiting spectral measure. This
phenomenon can be fully described in terms of free probability involving the
subordination function related to the additive free convolution of the limiting
spectral measure of the perturbation matrix by a semi-circular distribution.
Note that up to now only finite rank perturbations had been considered (even in
the deformed GUE case)
The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and non-universality of the fluctuations
We investigate the asymptotic spectrum of deformed Wigner matrices. The deformation is deterministic will all but finitely many eigenvalues equal to zero. We show that, as soon as the first largest or last smallest eigenvalues of the deformation are sufficiently far from 0, the corresponding eigenvalues of the deformed Wigner matrix almost surely exit the limiting semicircle compact support as the size of the matrix becomes large. In the particular case of a diagonal pertubation of rank 1, we prove that the fluctuations of the largest eigenvalue are not universal and depend on the particular distribution of the entries of the Wigner matrix
OUTLIERS IN THE SPECTRUM OF LARGE DEFORMED UNITARILY INVARIANT MODELS
44 pages, LaTeX, one figure. Several changes form the previous version, including a correction in Lemma 3.1 and the addition of several remarks and of a numerical example, kindly provided by Charles Bordenave. Same title as http://arxiv.org/abs/1207.5443, which it greatly expands.International audienceIn 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
Random matrices and random graphs*
We collect recent results on random matrices and random graphs. The topics covered are: fluctuations of the empirical measure of random matrices, finite-size effects of algorithms involving random matrices, characteristic polynomial of sparse matrices and Voronoi tesselations of split trees