Séminaire de Probabilités commun ICJ/UMPA

On the enumeration of maps with three tight boundaries and its probabilistic consequences

by Grégory Miermont

Amphi A (UMPA)

Amphi A



We consider the enumeration problem of graphs on surfaces, or maps, with three boundaries, also colloquially called pairs of pants. Perhaps surprisingly, a formula due to Eynard and extended by Collet-Fusy shows that this problem has a very simple and explicit solution, which becomes even simpler when one asks that the boundaries are tight, meaning that they have smallest possible length in their free homotopy class. We provide a bijective approach to this formula which consists in decomposing the graph into elementary pieces in a way that is reminiscent of certain geometric constructions of pairs of pants in hyperbolic geometry. I will also discuss the probabilistic consequences of this bijective approach by studying statistics of minimal separating loops in random maps with boundaries. Based on joint work with Jérémie Bouttier and Emmanuel Guitter.