Multi-ended Markovian triangulations and robust convergence to the UIPT
Annales Henri Lebesgue, Volume 5 (2022), pp. 1235-1259.

KeywordsRandom planar maps, UIPT, spatial Markov property, pattern occurences

### Abstract

We classify completely the infinite, planar triangulations satisfying a weak spatial Markov property, without assuming one-endedness nor finiteness of vertex degrees. In particular, the Uniform Infinite Planar Triangulation (UIPT) is the only such triangulation with average degree $6$. As a consequence, we prove that the convergence of uniform triangulations of the sphere to the UIPT is robust, in the sense that it is preserved under various perturbations of the uniform measure. As another application, we obtain large deviation estimates for the number of occurencies of a pattern in uniform triangulations.

### References

[AMS20] Albenque, Marie; Ménard, Laurent; Schaeffer, Gilles Local convergence of large random triangulations coupled with an Ising model, Trans. Am. Math. Soc., Volume 374 (2020), p. 1 | DOI | MR | Zbl

[Ang03] Angel, Omer Growth and percolation on the uniform infinite planar triangulation, Geom. Funct. Anal., Volume 13 (2003) no. 5, pp. 935-974 | DOI | MR | Zbl

[AR15] Angel, Omer; Ray, Gourab Classification of half planar maps, Ann. Probab., Volume 43 (2015) no. 3, pp. 1315-1349 | DOI | MR | Zbl

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

[BL21] Budzinski, Thomas; Louf, Baptiste Local limits of uniform triangulations in high genus, Invent. Math., Volume 223 (2021), pp. 1-47 | DOI | MR | Zbl

[BL22] Budzinski, Thomas; Louf, Baptiste Local limits of bipartite maps with prescribed face degrees in high genus, Ann. Probab., Volume 50 (2022) no. 3, pp. 1059-1126 | DOI | MR | Zbl

[BS14] Björnberg, Jakob E.; Stefansson, Sigurdur Orn Recurrence of bipartite planar maps, Electron. J. Probab., Volume 19 (2014) no. 31, pp. 1-40 | MR | Zbl

[Bud18] Budzinski, Thomas The hyperbolic Brownian plane, Probab. Theory Relat. Fields, Volume 171 (2018) no. 1, pp. 503-541 | DOI | MR | Zbl

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

[Che] Chen, Linxiao personal communication

[CKM] Curien, Nicolas; Kortchemski, Igor; Marzouk, Cyril The mesoscopic geometry of sparse random maps (to appear in Journal de l’École Polytechnique – Mathématiques)

[CT20a] Chen, Linxiao; Turunen, Joonas Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits, Commun. Math. Phys., Volume 374 (2020) no. 3, pp. 1577-1643 | DOI | MR | Zbl

[CT20b] Chen, Linxiao; Turunen, Joonas Ising model on random triangulations of the disk: phase transition (2020) (https://arxiv.org/abs/2003.09343v1)

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

[Cur19] Curien, Nicolas Peeling random planar maps, Saint-Flour lecture notes (2019)

[CV81] Cori, Robert; Vauquelin, Bernard Planar maps are well labeled trees, Can. J. Math., Volume 33 (1981) no. 5, pp. 1023-1042 | DOI | MR | Zbl

[DS20] Drmota, Michael; Stufler, Benedikt Pattern occurrences in random planar maps, Stat. Probab. Lett., Volume 158 (2020), 108666 | DOI | MR | Zbl

[GJ08] Goulden, Ian P.; Jackson, David M. The KP hierarchy, branched covers, and triangulations, Adv. Math., Volume 219 (2008) no. 3, pp. 932-951 | DOI | MR | Zbl

[HS33] Hildebrandt, Theophil H.; Schoenberg, Isaac J. On Linear Functional Operations and the Moment Problem for a Finite Interval in One or Several Dimensions, Ann. Math., Volume 34 (1933) no. 2, pp. 317-328 | DOI | MR | Zbl

[Kri07] Krikun, Maxim Explicit enumeration of triangulations with multiple boundaries, Electron. J. Comb., Volume 14 (2007) no. 1, R61 | MR | Zbl

[MM07] Marckert, Jean-François; Miermont, Grégory Invariance principles for random bipartite planar maps, Ann. Probab., Volume 35 (2007) no. 5, pp. 1642-1705 | MR | Zbl

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

[Tur20] Turunen, Joonas Interfaces in the vertex-decorated Ising model on random triangulations of the disk (2020) (https://arxiv.org/abs/2003.11012)

[Tut62] Tutte, William T. A census of planar triangulations, Can. J. Math., Volume 14 (1962), pp. 21-38 | DOI | MR | Zbl