Cumulants in Noncommutative Probability Theory IV. De Finetti's Theorem and LpL^p-Inequalities

In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. The main result is a free version of De Finetti's theorem, characterising amalgamated free products as noncrossing exchangeability systems which satisfy a so-called weak singleton condition. The main tool in the proof is an $L^p$-inequality with uniformly bounded constants for i.i.d. sequences in noncrossing exchangeability systems.Comment: 33 pages, AMS LaTeX; Brillinger's formula transferred to separate pape

Cumulants in noncommutative probability II. Generalized Gaussian random variables

We continue the investigation of noncommutative cumulants. In this paper various characterizations of noncommutative Gaussian random variables are proved.Comment: 14 pages, AMS-LaTeX; many corrections according to the referee; to appear in Prob Th Rel Field

Cumulants, lattice paths, and orthogonal polynomials

A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.Comment: 11 pages, AMS LaTeX, uses pstricks; revised according to referee's suggestions, in particular cut down last section and corrected some wrong attribution

On the computation of spectra in free probability

We use free probability techniques to compute borders of spectra of non hermitian operators in finite von Neumann algebras which arise as `free sums' of `simple' operators. To this end, the resolvent is analyzed with the aid of the Haagerup inequality. Concrete examples coming from reduced C*-algebras of free product groups and leading to systems of polynomial equations illustrate the approach.Comment: LaTeX2e, 15 pages, 3 figure

On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs

We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete orthonormal basis of finitely supported eigenfunctions.Comment: notational improvements and minor corrections, 6 pages; complement to arXiv:0712.313

Free Lamplighter Groups and a Question of Atiyah

We compute the von Neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational, thus providing an elementary constructive answer to a question of Atiyah.Comment: AMSLaTeX, 10 pages, to appear in Amer J Mat

Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series

We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and M\"obius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faa di Bruno's theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell-Baker-Hausdorff series in this context. In a final section we exhibit an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients.Comment: some minor corrections, 48 page