Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants

Bouttier, Jérémie; Guitter, Emmanuel; Miermont, Grégory

Annales Henri Lebesgue, Tome 5 (2022), pp. 1035-1110.

Nous considérons des cartes planaires à trois bords, plus familièrement appelées « pantalons ». Dans le cas de cartes biparties avec un contrôle sur le degré des faces, une expression simple pour leur fonction génératrice a été trouvée par Eynard et prouvée bijectivement par Collet et Fusy. Dans cet article nous obtenons une formule encore plus simple pour les pantalons serrés, c’est-à-dire pour les cartes dont les bords ont une longueur minimale dans leur classe d’homotopie. Nous utilisons une approche bijective basée sur la décomposition en tranches, que nous étendons en introduisant de nouveaux blocs fondamentaux appelés triangles et diangles bigéodésiques, et en travaillant sur le revêtement universel de la sphère à trois trous. Nous discutons également la statistique des longueurs minimales des boucles séparantes au sein des pantalons et des cartes annulaires (non nécessairement serrés), et leur asymptotique dans la limite des grandes tailles.

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for tight pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit.

Classification : 05A15, 05A19, 60C05, 60F05
Mots clés : Planar maps with three boundaries, generating functions, bijective enumeration, slice decomposition

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