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)

Infinitesimal non-crossing cumulants and free probability of type B

Free probabilistic considerations of type B first appeared in a paper by Biane, Goodman and Nica in 2003. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko (arXiv:0903.2721). The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A_1, ..., A_k in an infinitesimal noncommutative probability space (A, phi, phi'), and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, phi, phi'), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A_1, ... A_k is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from "usual" free probability. We show that the lattices of non-crossing partitions of type B appear in the combinatorial study of (A, phi, phi'), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A_1, ..., A_k in (A, phi) can be naturally upgraded to infinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companion functional" phi'.Comment: 38 pages, 1 figur

Liberté infinitésimale et modèles matriciels déformés

Le travail effectué dans cette thèse concerne les domaines de la théorie des matrices aléatoires et des probabilités libres, dont on connaît les riches connexions depuis le début des années 90. Les résultats s'organisent principalement en deux parties : la première porte sur la liberté infinitésimale, la seconde sur les matrices aléatoires déformées. Plus précisément, on jette les bases d'une théorie combinatoire de la liberté infinitésimale, au premier ordre d'abord, telle que récemment introduite par Belinschi et Shlyakhtenko, puis aux ordres supérieurs. On en donne un cadre simple et général, et on introduit des fonctionnelles de cumulants non-croisés, caractérisant la liberté infinitésimale. L'accent est mis sur la combinatoire et les idées d'essence différentielle qui sous-tendent cette notion. La seconde partie poursuit l'étude des déformations de modèles matriciels, qui a été ces dernières années un champ de recherche très actif. Les résultats présentés sont originaux en ce qu'ils concernent des perturbations déterministes Hermitiennes de rang non nécessairement fini de matrices de Wigner et de Wishart. En outre, un apport de ce travail est la mise en lumière du lien entre la convergence des valeurs propres de ces modèles et les probabilités libres, plus particulièrement le phénomène de subordination pour la convolution libre. Ce lien donne une illustration de la puissance des idées des probabilités libres dans les problèmes de matrices aléatoires.This thesis is about Random Matrix Theory and Free Probability whose strong relation is known since the early nineties. The results mainly organize in two parts : one on infinitesimal freeness, the other on deformed matrix models.More precisely, a combinatorial theory of first order infinitesimal freeness, as introduced by Belinschi and Shlyakhtenko, is developed and generalized to higher order. We give a simple and general framework and we introduce infinitesimal non-crossing cumulant functionals, providing a characterization of infinitesimal freeness. The emphasis is put on combinatorics and on the essentially differential ideas underlying this notion. The second part carries further the study of deformations of matrix models, which has been a very active field of research these past years. The results we present are original in the sense they deal with non-necessarily finite rank deterministic Hermitian perturbations of Wigner and Wishart matrices. Moreover, these results shed light on the link between convergence of eigenvalues of deformed matrix models and free probability, particularly the subordination phenomenon related to free convolution. This link gives an illustration of the power of free probability ideas in random matrix problems

3D multiphoton characterization of χ(2) nonlinearity induced in a multimode fiber through optical poling

We characterize the 3D spatial distribution of the quadratic susceptibility in an optically poled graded-index multimode fiber in presence of spatial beam self-cleaning. We also show how the poling process can improve the beam self-cleaning

Towards a new understanding of optical poling efficiency in multimode fibers

International audienceAll-optical poling was demonstrated for the first time in 1986 in single mode fibers: such nonlinear optical process enabled the introduction of a second-order susceptibility (χ (2)) in a doped silica fiber. By simply using an intense laser source, alloptical poling, later theoretically described by Stolen and coworkers, permitted the generation of a second harmonic (SH) signal in an otherwise centrosymmetric doped material. More recently, similar experiments have been carried out by exploiting complex beam propagation in multimode fibers. In this work we reveal, for the first time to our knowledge, the 3D spatial distribution of a χ (2) nonlinearity written in a graded-index (GRIN) multimode (MM) fiber. In particular, the presence of a doubly-periodic distribution of χ (2) is unveiled by means of multiphoton microscopy. The shortest period (tens of micrometers) is due to the beating between the fundamental and the SH beams, and it is responsible for their quasiphase matching (QPM). Whereas the longest period (hundreds of micrometers) is associated with the periodic evolution, or self-imaging, of the power density of the MM beam along the GRIN MM fiber. The complex modal beating, leading to spatial self-cleaning of the fundamental beam, is thus printed inside the fiber core, and revealed by our measurements. We considered two fibers of similar composition and opto-geometric parameters, and we compared the evolution of the optical poling process with time. Despite the rather similar fiber characteristics, we observed a striking difference in the poling efficiency between the two fibers. Such observation led us to point out the importance of considering the complete fiber fabrication process (both the preform elaboration and the drawing steps) on the final structure and microstructure of optical fibers