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

Impact of the drought on the fodder self-sufficiency of organic and conventional highland dairy farms

Eight highland dairy farms in the French Massif Central (4 organic and 4 conventional) were surveyed from 2000 to 2005 to understand the forage system functioning and the specificities of organic farms. During this period two important droughts occurred, which highly affected the fodder self-sufficiency of the organic farms, having consequences on more than a year of production. The conventional farms were less affected than the organic ones, and the farmers developed varied strategies including a reduction of the LU and the use of more maize. To maintain the stability of the milk production, organic farms had to increase the reliance on external fodder resources. The lack of security forage stores can explain the sensitivity of these farms and their incapacity to recover a good level of self-sufficiency

The astronomical units

The IAU-1976 System of astronomical constants includes three astronomical units (i.e. for time, mass and length). This paper reports on the status of the astronomical unit of length (ua) and mass (MSun) within the context of the recent IAU Resolutions on reference systems and the use of modern observations in the solar system. We especially look at a possible re-definition of the ua as an astronomical unit of length defined trough a fixed relation to the SI metre by a defining number.Comment: 2 pages, to be published in the Proceedings of the "Journees 2008 Systemes de reference spatio-temporels

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 $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincar\'{e} inequality. The matrix $A_N$ 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 $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semicircle compact support as the size $N$ becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of $W_N$. On the other hand, when $A_N$ 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 $W_N$.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)