Extended modular operad

This paper is a sequel to [LoMa] where moduli spaces of painted stable curves were introduced and studied. We define the extended modular operad of genus zero, algebras over this operad, and study the formal differential geometric structures related to these algebras: pencils of flat connections and Frobenius manifolds without metric. We focus here on the combinatorial aspects of the picture. Algebraic geometric aspects are treated in [Ma2].Comment: 38 pp., amstex file, no figures. This version contains additional references and minor change

Arthur Parameters and Fourier coefficients for Automorphic Forms on Symplectic Groups

We study the structures of Fourier coefficients of automorphic forms on symplectic groups based on their local and global structures related to Arthur parameters. This is a first step towards the general conjecture on the relation between the structure of Fourier coefficients and Arthur parameters given by the first named author in [J14].Comment: Final version. 44 pages. Minor revisions. To appear in the Annales de l'Institut Fourie

Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees

We consider a family of random trees satisfying a Markov branching property. Roughly, this property says that the subtrees above some given height are independent with a law that depends only on their total size, the latter being either the number of leaves or vertices. Such families are parameterized by sequences of distributions on partitions of the integers that determine how the size of a tree is distributed in its different subtrees. Under some natural assumption on these distributions, stipulating that "macroscopic" splitting events are rare, we show that Markov branching trees admit the so-called self-similar fragmentation trees as scaling limits in the Gromov-Hausdorff-Prokhorov topology. The main application of these results is that the scaling limit of random uniform unordered trees is the Brownian continuum random tree. This extends a result by Marckert-Miermont and fully proves a conjecture by Aldous. We also recover, and occasionally extend, results on scaling limits of consistent Markov branching models and known convergence results of Galton-Watson trees toward the Brownian and stable continuum random trees.Comment: Published in at http://dx.doi.org/10.1214/11-AOP686 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.Comment: 37 pages, 24 figure