Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants
Annales Henri Lebesgue, Volume 5 (2022), pp. 1035-1110.

KeywordsPlanar maps with three boundaries, generating functions, bijective enumeration, slice decomposition

### Abstract

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.

### References

[AB12] Albenque, Marie; Bouttier, Jérémie Constellations and multicontinued fractions: application to Eulerian triangulations, Proceedings of the 24th international conference on formal power series and algebraic combinatorics, FPSAC 2012 (Discrete Mathematics and Theoretical Computer Science), Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS), 2012, pp. 805-816 | DOI | MR | Zbl

[AB13] Ambjørn, Jan; Budd, Timothy G. Trees and spatial topology change in causal dynamical triangulations, J. Phys. A, Math. Gen., Volume 46 (2013) no. 31, 315201 | DOI | MR | Zbl

[ADJ97] Ambjørn, Jan; Durhuus, Bergfinnur; Jonsson, Thordur Quantum geometry. A statistical field theory approach, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 1997 | DOI | MR | Zbl

[AJM90] Ambjørn, Jan; Jurkiewicz, Jerzy; Makeenko, Yu. M. Multiloop correlators for two-dimensional quantum gravity, Phys. Lett., B, Volume 251 (1990) no. 4, pp. 517-524 | DOI | MR

[BBI01] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001 | DOI | MR | Zbl

[BDFG04] Bouttier, Jérémie; Di Francesco, Philippe; Guitter, Emmanuel Planar maps as labeled mobiles, Electron. J. Comb., Volume 11 (2004) no. 1, 69 | MR | Zbl

[Bet16] Bettinelli, Jérémie Geodesics in Brownian surfaces (Brownian maps), Ann. Inst. Henri Poincaré, Probab. Stat., Volume 52 (2016) no. 2, pp. 612-646 | DOI | MR | Zbl

[BF12] Bernardi, Olivier; Fusy, Éric Unified bijections for maps with prescribed degrees and girth, J. Comb. Theory, Ser. A, Volume 119 (2012) no. 6, pp. 1351-1387 | DOI | MR | Zbl

[BFG14] Bouttier, Jérémie; Fusy, Éric; Guitter, Emmanuel On the two-point function of general planar maps and hypermaps, Ann. Inst. Henri Poincaré D, Comb. Phys. Interact., Volume 1 (2014) no. 3, pp. 265-306 | DOI | MR | Zbl

[BG08] Bouttier, Jérémie; Guitter, Emmanuel The three-point function of planar quadrangulations, J. Stat. Mech. Theory Exp. (2008) no. 7, P07020 | DOI | MR | Zbl

[BG09a] Bouttier, Jérémie; Guitter, Emmanuel Confluence of geodesic paths and separating loops in large planar quadrangulations, J. Stat. Mech. Theory Exp. (2009) no. 3, P03001 | DOI | MR | Zbl

[BG09b] Bouttier, Jérémie; Guitter, Emmanuel Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop, J. Phys. A, Math. Gen., Volume 42 (2009) no. 46, 465208 | DOI | MR | Zbl

[BG12] Bouttier, Jérémie; Guitter, Emmanuel Planar maps and continued fractions, Commun. Math. Phys., Volume 309 (2012) no. 3, pp. 623-662 | DOI | MR | Zbl

[BG14] Bouttier, Jérémie; Guitter, Emmanuel On irreducible maps and slices, Comb. Probab. Comput., Volume 23 (2014) no. 6, pp. 914-972 | DOI | MR | Zbl

[BM17] Bettinelli, Jérémie; Miermont, Grégory Compact Brownian surfaces I: Brownian disks, Probab. Theory Relat. Fields, Volume 167 (2017) no. 3-4, pp. 555-614 | DOI | MR | Zbl

[BM22] Bettinelli, Jérémie; Miermont, Grégory Compact Brownian surfaces II: Orientable surfaces, 2022+ (In preparation)

[BMR19] Baur, Erich; Miermont, Grégory; Ray, Gourab Classification of scaling limits of uniform quadrangulations with a boundary, Ann. Probab., Volume 47 (2019) no. 6, pp. 3397-3477 | DOI | MR | Zbl

[Bou19] Bouttier, Jérémie Planar maps and random partitions (2019) (Habilitation à diriger des recherches, Université Paris-Sud, https://arxiv.org/abs/1912.06855)

[Bud17] Budd, Timothy G. Peeling of random planar maps, 2017 (Lecture notes for a mini-course given at the Mini-school on Random Maps and the Gaussian Free Field, ENS de Lyon, https://hef.ru.nl/~tbudd/docs/mappeeling.pdf)

[Bud22a] Budd, Timothy G. Irreducible metric maps and Weil–Petersson volumes, Commun. Math. Phys., Volume 394 (2022), pp. 887-917 | DOI | MR | Zbl

[Bud22b] Budd, Timothy G. On polynomials counting essentially irreducible maps, Electron. J. Comb., Volume 29 (2022) no. 2, P2.45 | DOI | MR | Zbl

[CdVE10] Colin de Verdière, Éric; Erickson, Jeff Tightening nonsimple paths and cycles on surfaces, SIAM J. Comput., Volume 39 (2010) no. 8, pp. 3784-3813 | DOI | MR | Zbl

[CdVL07] Colin de Verdière, Éric; Lazarus, Francis Optimal pants decompositions and shortest homotopic cycles on an orientable surface, J. ACM, Volume 54 (2007) no. 4, 18 | DOI | MR | Zbl

[CF12] Collet, Gwendal; Fusy, Éric A simple formula for the series of bipartite and quasi-bipartite maps with boundaries, Proceedings of the 24th international conference on formal power series and algebraic combinatorics, FPSAC 2012 (Discrete Mathematics and Theoretical Computer Science), Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS), 2012, pp. 607-618 | MR | Zbl

[Cur19] Curien, Nicolas Peeling random planar maps, 2019 (Lecture notes for the Saint-Flour Probability Summer School, https://www.math.u-psud.fr/~curien/enseignement.html)

[DFGZJ95] Di Francesco, Philippe; Ginsparg, Paul; Zinn-Justin, Jean $2$D gravity and random matrices, Phys. Rep., Volume 254 (1995) no. 1-2, p. 133 | DOI | MR

[Eyn16] Eynard, Bertrand Counting surfaces. CRM Aisenstadt chair lectures, Progress in Mathematical Physics, 70, Birkhäuser, 2016 | DOI | MR | Zbl

[FG14] Fusy, Éric; Guitter, Emmanuel The three-point function of general planar maps, J. Stat. Mech. Theory Exp. (2014) no. 9, P09012 | DOI | MR | Zbl

[IT92] Imayoshi, Yoichi; Taniguchi, Masahiko An introduction to Teichmüller spaces, Springer, 1992 (Translated and revised from the Japanese by the authors) | DOI | MR | Zbl

[LG13] Le Gall, Jean-François Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | DOI | MR | Zbl

[LGM12] Le Gall, Jean-François; Miermont, Grégory Scaling limits of random trees and planar maps, Probability and statistical physics in two and more dimensions (Clay Mathematics Proceedings), Volume 15, American Mathematical Society, 2012, pp. 155-211 | MR | Zbl

[Mie09] Miermont, Grégory Tessellations of random maps of arbitrary genus, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 5, pp. 725-781 | DOI | Numdam | MR | Zbl

[Mie14] Miermont, Grégory Aspects of random maps, 2014 (Lecture notes for the Saint-Flour Probability Summer School, http://perso.ens-lyon.fr/gregory.miermont/coursSaint-Flour.pdf)

[Mil18] Miller, Jason, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures (2018), pp. 2945-2971 | MR | Zbl

[MN16] Marques, Fernando C.; Neves, André Topology of the space of cycles and existence of minimal varieties, Advances in geometry and mathematical physics. Lectures given at the geometry and topology conference at Harvard University (Surveys in Differential Geometry), Volume 21, International Press, 2016, pp. 165-177 | MR | Zbl

[Nor10] Norbury, Paul Counting lattice points in the moduli space of curves, Math. Res. Lett., Volume 17 (2010) no. 3, pp. 467-481 | DOI | MR | Zbl

[Rie22] Riera, Armand Isoperimetric inequalities in the Brownian plane, Ann. Probab., Volume 50 (2022) no. 5, pp. 2013-2055 | MR

[Sch15] Schaeffer, Gilles Planar maps, Handbook of enumerative combinatorics (Bóna, Miklós, ed.) (Discrete Mathematics and its Applications), CRC Press, 2015, pp. 335-395 | DOI | MR | Zbl

[Sti92] Stillwell, John C. Geometry of Surfaces, Universitext, Springer, 1992 | DOI | Zbl

[Thu97] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, 35, Princeton University Press, 1997 (edited by Silvio Levy) | DOI | MR | Zbl

[Tut63] Tutte, William T. A census of planar maps, Can. J. Math., Volume 15 (1963), pp. 249-271 | DOI | MR | Zbl

[Tut68] Tutte, William T. On the enumeration of planar maps, Bull. Am. Math. Soc., Volume 74 (1968), pp. 64-74 | DOI | MR | Zbl