Volume 1 (2018)
Curien, Nicolas;  Ménard, Laurent
The skeleton of the UIPT, seen from infinity
Annales Henri Lebesgue, Volume 1 (2018), pp. 87-125.

Metadata

DOI10.5802/ahl.3
Keywords random planar maps, UIPT, skeleton decomposition, geodesic confluence, discrete $3/2$ stable trees

Abstract

We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction measuring distances from infinity in the UIPT. We then use this horofunction to define the skeleton “seen from infinity” of the UIPT and relate it to a simple Galton–Watson tree conditioned to survive, giving a new and particularly simple construction of the UIPT. Scaling limits of perimeters and volumes of horohulls within this new decomposition are also derived, as well as a new proof of the two point function formula for random triangulations in the scaling limit due to Ambjørn and Watabiki.


References

[AB16] Ambjørn, J.; Budd, T. Multi-point functions of weighted cubic maps, Ann. Inst. H. Poincaré D, Volume 3 (2016), pp. 1-44 | DOI | MR | Zbl

[ADG18] Abraham, Romain; Delmas, Jean-François; Guo, Hongsong Critical Multi-Type Galton-Watson Trees Conditioned to be Large (2018), pp. 757-788 | Zbl

[ADJ97] Ambjørn, Jan; Durhuus, Bergfinnur; Jonsson, Thordur Quantum geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997, xiv+363 pages (A statistical field theory approach) | DOI | Zbl

[AR18] Angel, Omer; Ray, Gourab The half plane UIPT is recurrent, Probab. Theory Related Fields, Volume 170 (2018) no. 3-4, pp. 657-683 | DOI | MR | Zbl

[AS03] Angel, Omer; Schramm, Oded Uniform infinite planar triangulations, Comm. Math. Phys., Volume 241 (2003) no. 2-3, pp. 191-213 | DOI | MR | Zbl

[AW95] Ambjørn, J.; Watabiki, Y. Scaling in quantum gravity, Nuclear Phys. B, Volume 445 (1995) no. 1, pp. 129-142 | DOI | MR | Zbl

[BDFG03] Bouttier, J.; Di Francesco, P.; Guitter, E. Geodesic distance in planar graphs, Nuclear Phys. B, Volume 663 (2003) no. 3, pp. 535-567 | DOI | MR | Zbl

[BDFG04] Bouttier, J.; Di Francesco, P.; Guitter, E. Planar maps as labeled mobiles, Electron. J. Combin., Volume 11 (2004) no. 1, Research Paper 69, 27 pages | DOI | 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

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

[Bud18] Budzinski, T. Infinite geodesics in hyperbolic random triangulations (2018) (arXiv:1804.07711)

[CD06] Chassaing, Philippe; Durhuus, Bergfinnur Local limit of labeled trees and expected volume growth in a random quadrangulation, Ann. Probab., Volume 34 (2006) no. 3, pp. 879-917 | DOI | MR | Zbl

[CLG16] Curien, Nicolas; Le Gall, Jean-François The hull process of the Brownian plane, Probab. Theory Related Fields, Volume 166 (2016) no. 1-2, pp. 187-231 | DOI | MR | Zbl

[CLG18] Curien, Nicolas; Le Gall, Jean-François First passage percolation and local modifications of distances in random triangulations (2018) (to appear in Ann. Sci. ENS) | Zbl

[CM] Curien, N.; Marzouk, C. Random walks on Boltzmann planar maps are always sub-diffusive (in Preparation)

[CMM13] Curien, N.; Ménard, L.; Miermont, G. A view from infinity of the uniform infinite planar quadrangulation, ALEA Lat. Am. J. Probab. Math. Stat., Volume 10 (2013) no. 1, pp. 45-88 | MR | Zbl

[Cur16] Curien, Nicolas Planar stochastic hyperbolic triangulations, Probab. Theory Related Fields, Volume 165 (2016) no. 3-4, pp. 509-540 | DOI | MR | Zbl

[DF05] Di Francesco, P. Geodesic distance in planar graphs: an integrable approach, Ramanujan J., Volume 10 (2005) no. 2, pp. 153-186 | DOI | MR | Zbl

[DLG02] Duquesne, Thomas; Le Gall, Jean-François Random trees, Lévy processes and spatial branching processes, Astérisque, Soc. Math. France, 2002 no. 281, vi+147 pages | Numdam | Zbl

[FS09] Flajolet, Philippe; Sedgewick, Robert Analytic combinatorics, Cambridge University Press, Cambridge, 2009, xiv+810 pages | Zbl

[GJ04] Goulden, Ian P.; Jackson, David M. Combinatorial enumeration, Dover Publications, Inc., Mineola, NY, 2004, xxvi+569 pages (With a foreword by Gian-Carlo Rota; Reprint of the 1983 original) | Zbl

[Gui17] Guitter, Emmanuel The distance-dependent two-point function of triangulations: a new derivation from old results, Ann. Inst. Henri Poincaré D, Volume 4 (2017) no. 2, pp. 177-211 | DOI | MR | Zbl

[Jeu82] Jeulin, T. Sur la convergence absolue de certaines intégrales, Seminar on Probability, XVI (Lecture Notes in Math.), Volume 920, Springer, Berlin-New York (1982), pp. 248-256 | Numdam | MR | Zbl

[Kri04] Krikun, M. A. A uniformly distributed infinite planar triangulation and a related branching process, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 307 (2004) no. 10, p. 141-174, 282–283 | Zbl

[LG10] Le Gall, Jean-François Geodesics in large planar maps and in the Brownian map, Acta Math., Volume 205 (2010) no. 2, pp. 287-360 | MR | Zbl

[LGL17] Le Gall, Jean-François; Lehéricy, Thomas Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation (2017) (to appear in Ann. Probab.) | arXiv | Zbl

[Mén18] Ménard, Laurent Volumes in the Uniform Infinite Planar Triangulation:from skeletons to generating functions, Combin. Probab. and Comp., Volume 27 (2018) no. 6, pp. 946-973 | DOI | MR | Zbl

[Ste18] Stephenson, Robin Local convergence of large critical multi-type Galton-Watson trees and applications to random maps, J. Theoret. Probab., Volume 31 (2018) no. 1, pp. 159-205 | DOI | MR | Zbl