Invited Talks

Bénédicte Haas, University of Paris-Dauphine, France Limits of Non-increasing Markov Chains and Applications to Random Trees and Coalescents Consider a non-increasing Markov chain with values in the set of non-negative integers, starting from a large integer !. We describe its scaling limit as ! → ∞, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. This result entails various applications, in particular to the description of scaling limits of a large set of sequences of random trees satisfying a branching property (Markov branching trees). For example, from this, we can prove that a tree uniformly distributed among the set of rooted un-ordered unlabeled trees with ! vertices has the so-called Brownian continuum random tree as scaling limit. We can also recover the well-known results by Aldous and Duquesne on large conditioned critical Galton-Watson trees that converge to the Brownian continuum random tree or to the stable Lévy trees. Last, other applications of the Markov chains scaling limits will be discussed, for example to the number of collisions in certain coalescent processes !-coalescents starting with a large number of particles.