Bénédicte Haas

Loss of mass in deterministic and random fragmentations

We consider a linear rate equation, depending on three parameters, that model fragmentation. For each of these fragmentation equations, there is a corresponding stochastic model, from which we construct an explicit solution to the equation. This solution is proved unique. We then use this solution to obtain criteria for the presence or absence of loss of mass in the fragmentation equation, as a function of the equation parameters. Next, we investigate small and large times asymptotic behavior of the total mass for a wide class of parameters. Finally, we study the loss of mass in the stochastic models.

Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldouss beta-splitting models and Fords alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.

Spinal partitions and invariance under re-rooting of continuum random trees

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson-Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting.

Self-similar scaling limits of non-increasing Markov chains

We study scaling limits of non-increasing Markov chains with values in the set of non-negative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately rescaled, converges in distribution, as n → ∞, to a non-increasing self-similar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the self-similar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in Λ-coalescents that do not descend from infinity and of non-consistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in the forthcoming paper [1 1].

Behavior near the extinction time in self-similar fragmentations I: The stable case

The stable fragmentation with index of self-similarity

Scaling limits of k-ary growing trees

\alpha \in [-1/2,0)

Equilibrium for fragmentation with immigration

is derived by looking at the masses of the subtrees formed by discarding the parts of a

Asymptotics of heights in random trees constructed by aggregation

(1 + \alpha)^{-1}

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

--stable continuum random tree below height

The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree

t