Flows on rooted trees and the Narayana idempotents

Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the Pre-Lie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the group of series associated with the Dendriform operad. This allows to recover the Lie idempotents in the descent algebras recently introduced by Menous, Novelli and Thibon. Moreover, one defines new Lie idempotents and conjecture the existence of some others.Comment: 37 pages ; 6 figure

Limit theorems for linear eigenvalue statistics of overlapping matrices

The paper proves several limit theorems for linear eigenvalue statistics of overlapping Wigner and sample covariance matrices. It is shown that the covariance of the limiting multivariate Gaussian distribution is diagonalized by choosing the Chebyshev polynomials of the first kind as the basis for the test function space. The covariance of linear statistics for the Chebyshev polynomials of sufficiently high degree depends only on the first two moments of the matrix entries. Proofs are based on a graph-theoretic interpretation of the Chebyshev linear statistics as sums over non-backtracking cyclic pathsComment: 44 pages, 4 figures, some typos are corrected and proofs clarified. Accepted to the Electronic Journal of Probabilit