Graph limits of random graphs from a subset of connected k-trees.

For any set Ω of non-negative integers such that { 0 , 1 } ⊊ Ω , we consider a random Ω-k-tree G n,k that is uniformly selected from all connected k-trees of (n + k) vertices such that the number of (k + 1)-cliques that contain any fixed k-clique belongs to Ω. We prove that Gn,k, scaled by ( k H k σ Ω ) / ( 2 n ) where H k is the kth harmonic number and σ Ω > 0, converges to the continuum random tree T e . Furthermore, we prove local convergence of the random Ω-k-tree G n , k ∘ to an infinite but locally finite random Ω-k-tree G∞,k.

Stochastic analysis of the extra clustering model for animal grouping.

We consider the extra clustering model which was introduced by Durand et al. (J Theor Biol 249(2):262-270, 2007) in order to describe the grouping of social animals and to test whether genetic relatedness is the main driving force behind the group formation process. Durand and François (J Math Biol 60(3):451-468, 2010) provided a first stochastic analysis of this model by deriving (amongst other things) asymptotic expansions for the mean value of the number of groups. In this paper, we will give a much finer analysis of the number of groups. More precisely, we will derive asymptotic expansions for all higher moments and give a complete characterization of the possible limit laws. In the most interesting case (neutral model), we will prove a central limit theorem with a surprising normalization. In the remaining cases, the limit law will be either a mixture of a discrete and continuous law or a discrete law. Our results show that, except of in degenerate cases, strong concentration around the mean value takes place only for the neutral model, whereas in the remaining cases there is also mass concentration away from the mean.

Embedded trees and the support of the ISE.

Embedded trees are labelled rooted trees, where the root has zero label and where the labels of adjacent vertices differ (at most) by [Formula: see text]. Recently it has been proved (see Chassaing and Schaeffer (2004) [8] and Janson and Marckert (2005) [11]) that the distribution of the maximum and minimum labels are closely related to the support of the density of the integrated superbrownian excursion (ISE). The purpose of this paper is to make this probabilistic limiting relation more explicit by using a generating function approach due to Bouttier et al. (2003) [6] that is based on properties of Jacobi's [Formula: see text]-functions. In particular, we derive an integral representation of the joint distribution function of the supremum and infimum of the support of the ISE in terms of the Weierstrass [Formula: see text]-function. Furthermore we re-derive the limiting radius distribution in random quadrangulations (by Chassaing and Schaeffer (2004) [8]) with the help of exact counting generating functions.