Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed or localized vectors

The celebrated Marchenko-Pastur theorem gives the asymptotic spectral distribution of sums of random, independent, rank-one projections. Its main hypothesis is that these projections are more or less uniformly distributed on the first grassmannian, which implies for example that the corresponding vectors are delocalized, i.e. are essentially supported by the whole canonical basis. In this paper, we propose a way to drop this delocalization assumption and we generalize this theorem to a quite general framework, including random projections whose corresponding vectors are localized, i.e. with some components much larger than the other ones. The first of our two main examples is given by heavy tailed random vectors (as in a model introduced by Ben Arous and Guionnet or as in a model introduced by Zakharevich where the moments grow very fast as the dimension grows). Our second main example is given by vectors which are distributed as the Brownian motion on the unit sphere, with localized initial law. Our framework is in fact general enough to get new correspondences between classical infinitely divisible laws and some limit spectral distributions of random matrices, generalizing the so-called Bercovici-Pata bijection.Comment: 40 pages, 10 figures, some minor mistakes correcte

A matrix interpolation between classical and free max operations: I. The univariate case

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.Comment: 14 page

Liberation of orthogonal Lie groups

We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: $O_n,S_n,H_n,B_n,S_n',B_n'$. We investigate the representation theory aspects of the correspondence, with the result that for $O_n,S_n,H_n,B_n$, this is compatible with the Bercovici-Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.Comment: 42 page

Global spectrum fluctuations for the β\beta-Hermite and β\beta-Laguerre ensembles via matrix models

We study the global spectrum fluctuations for $\beta$-Hermite and $\beta$-Laguerre ensembles via the tridiagonal matrix models introduced in \cite{dumitriu02}, and prove that the fluctuations describe a Gaussian process on monomials. We extend our results to slightly larger classes of random matrices.Comment: 43 pages, 2 figures; typos correcte

On a class of free Levy laws related to a regression problem

The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.Comment: LaTeX, v2: strengthened main theore

Functional limit theorems for random regular graphs

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n. In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein's method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn-Szemer\'edi argument for estimating the second largest eigenvalue for all values of d and n.Comment: Added Remark 27. 39 pages. To appear in Probability Theory and Related Field

Adapted version of the Pubertal Development Scale for use in Brazil

OBJECTIVE: To determine whether scores in an adapted version of the self-assessment Pubertal Development Scale into Portuguese match those from the gold standard in pubertal development (Tanner scale). METHODS: This was a cross-sectional study with a convenience sample of 133 children and adolescents aged nine to 17 years (59 males; mean age of 13 years and six months, with standard deviation = 25 months). Youngsters completed the Pubertal Development Scale and were then examined by specialists in adolescent medicine. RESULTS: Exact absolute agreement of pubertal stages were modest, but significant associations between measures (correlation; intra-class correlation coefficients of consistency) showed that the Pubertal Development Scale adequately measures changes that map onto pubertal development determined by physical examination, on par with international publications. Furthermore, scores obtained from each Pubertal Development Scale question reflected adequate gonadal and adrenal events assessed by clinical ratings, mostly with medium/high effect sizes. Latent factors obtained from scores on all Pubertal Development Scale questions had excellent fit indices in Confirmatory Factor Analyses and correlated with Tanner staging. CONCLUSIONS: We conclude that self-assessment of body changes by youngsters using the Portuguese version of the Pubertal Development Scale is useful when estimates of pubertal progression are sufficient, and exact agreement with clinical staging is not necessary. The Pubertal Development Scale is, therefore, a reliable instrument for use in large-scale studies in Brazil that aim at investigating adolescent health related to pubertal developmental. The translated version and scoring systems are provided

Regioisomeric and substituent effects upon the outcome of the reaction of 1-borodienes with nitrosoarene compounds

A study of the reactivity of 1-borodienes with nitrosoarene compounds has been carried out showing an outcome that differs according to the hybridization state of the boron moiety. Using an sp2 boron substituent, a one-pot hetero-Diels–Alder/ring contraction cascade occurred to afford N-arylpyrroles with low to good yields depending on the electronic properties of the substituents on the borodiene, whereas an sp3 boron substituent led to the formation of stable boro-oxazines with high regioselectivity in most of the cases, in moderate to good yields. 1H and 11B NMR studies on two boro-oxazine regioisomers showed that selective deprotection can be performed. Formation of either the pyrrole or the furan derivative is pH- and regioisomer-structure-dependent. The results obtained, together with previous B3LYP calculations, support mechanistic proposals which suggest that pyrrole, or furan, formation proceeds via oxazine formation, followed by a boryl rearrangement and an intramolecular addition–elimination sequence