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Blossoming trees and the scaling limit of maps

Monday 20th April 2015 - 11:30 to 12:30
INI Seminar Room 1
Co-authors: Louigi Addario-Berry (Mc Gill University), Olivier Bernardi (Brandeis University), Gwendal Collet (TU Wien), Éric Fusy (CNRS), Dominique Poulalhon (Université Paris 7)

In the last years, numerous families of planar maps have been shown to converge to the Brownian map introduced by Miermont and Le Gall. Most of these results rely on some bijections with labeled trees mobiles due to Schaeffer and Bouttier, di Francesco and Guitter. In this talk, I'll present another class of bijections between so-called blossoming trees and maps. These bijections have been established 15 years ago but it is not since only recently that we managed to use them to track down the distances in the maps as a function of the trees. This link relies on some canonical « leftmost paths », which behave well both in the map and in the tree. As an example of the possible outcomes of these bijections, I'll prove that the scaling limit of simple maps (that is maps without loops nor multiple edges) is also the Brownian map. I'll emphasize the combinatorial construction which lies at the heart of this proof.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons