Convergence of non-bipartite maps via symmetrization of labeled trees
Annales Henri Lebesgue, Volume 4 (2021) , pp. 653-683.

KeywordsRandom trees, Invariance principle, Brownian snake, Random planar maps, Brownian map

### Abstract

Fix an odd integer $p\ge 5$. Let ${M}_{n}$ be a uniform $p$-angulation with $n$ vertices, endowed with the uniform probability measure on its vertices. We prove that there exists ${C}_{p}\in {ℝ}_{+}$ such that, after rescaling distances by ${C}_{p}/{n}^{1/4}$, ${M}_{n}$ converges in distribution for the Gromov–Hausdorff–Prokhorov topology towards the Brownian map. To prove the preceding fact, we introduce a bootstrapping principle for distributional convergence of random labelled plane trees. In particular, the latter allows to obtain an invariance principle for labeled multitype Galton–Watson trees, with only a weak assumption on the centering of label displacements.

### References

[ABA17] Addario-Berry, Louigi; Albenque, Marie The scaling limit of random simple triangulations and random simple quadrangulations, Ann. Probab., Volume 45 (2017) no. 5, pp. 2767-2825 | MR 3706731 | Zbl 1417.60022

[Abr16] Abraham, Céline Rescaled bipartite planar maps converge to the Brownian map, Ann. Inst. Henri Poincaré Probab. Stat., Volume 52 (2016) no. 2, pp. 575-595 | MR 3498001 | Zbl 1375.60034

[Ald91] Aldous, David The continuum random tree II: an overview, Stochastic analysis (Durham, 1990) (London Mathematical Society Lecture Note Series), Volume 167, Cambridge Univ. Press, Cambridge, 1991, pp. 23-70 | Article | MR 1166406 | Zbl 0791.60008

[BBI01] Burago, Dmiri; Burago, Yurĭi D.; Ivanov, Sergei V. A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001 | Article | MR 1835418 | Zbl 0981.51016

[BDG04] Bouttier, Jérémie; Di Francesco, Philippe; Guitter, Emmanuel Planar maps as labeled mobiles, Electron. J. Comb., Volume 11 (2004) no. 1, R69 | MR 2097335 | Zbl 1060.05045

[Bil13] Billingsley, Patrick Probability and measure. Anniversary edition, John Wiley & Sons, 2013 | Zbl 1236.60001

[BJM14] Bettinelli, Jérémie; Jacob, Emmanuel; Miermont, Grégory The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection, Electron. J. Probab., Volume 19 (2014), 74 | Zbl 1320.60088

[BLG13] Beltran, Johel; Le Gall, Jean-François Quadrangulations with no pendant vertices, Bernoulli, Volume 19 (2013) no. 4, pp. 1150-1175 | MR 3102547 | Zbl 1286.60003

[CLM13] Curien, Nicolas; Le Gall, Jean-François; Miermont, Grégory The Brownian cactus I. Scaling limits of discrete cactuses, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 2, pp. 340-373 | Article | Numdam | MR 3088373 | Zbl 1275.60035

[GPW09] Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees), Probab. Theory Relat. Fields, Volume 145 (2009) no. 1-2, pp. 285-322 | Article | MR 2520129 | Zbl 1215.05161

[LG05] Le Gall, Jean-François Random trees and applications, Probab. Surv., Volume 2 (2005), pp. 245-311 | MR 2203728 | Zbl 1189.60161

[LG07] Le Gall, Jean-François The topological structure of scaling limits of large planar maps, Invent. Math., Volume 169 (2007) no. 3, pp. 621-670 | Article | MR 2336042 | Zbl 1132.60013

[LG13] Le Gall, Jean-François Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | Article | MR 3112934 | Zbl 1282.60014

[Mar08] Marckert, Jean-François The lineage process in Galton–Watson trees and globally centered discrete snakes, Ann. Appl. Probab., Volume 18 (2008) no. 1, pp. 209-244 | MR 2380897 | Zbl 1140.60042

[Mar18] Marzouk, Cyril Scaling limits of random bipartite planar maps with a prescribed degree sequence, Random Struct. Algorithms, Volume 53 (2018) no. 3, pp. 448-503 | Article | MR 3854042 | Zbl 1397.05164

[Mie06] Miermont, Grégory An invariance principle for random planar maps, Fourth colloquium on mathematics and computer science IV. Algorithms, trees, combinatorics and probabilities. Papers based on the presentations at the colloquium, Nancy, France, September 18–22, 2006 (Discrete Mathematics and Theoretical Computer Science. Proceedings) (2006), pp. 39-58 | MR 2509622 | Zbl 1195.60049

[Mie08] Miermont, Grégory Invariance principles for spatial multitype Galton–Watson trees, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 44 (2008) no. 6, pp. 1128-1161 | Numdam | MR 2469338 | Zbl 1178.60058

[Mie09] Miermont, Grégory Tessellations of random maps of arbitrary genus, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 5, pp. 725-781 | Article | Numdam | MR 2571957 | Zbl 1228.05118

[Mie13] Miermont, Grégory The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., Volume 210 (2013) no. 2, pp. 319-401 | Article | MR 3070569 | Zbl 1278.60124

[MM03] Marckert, Jean-François; Mokkadem, Abdelkader States spaces of the snake and its tour: Convergence of the discrete snake, J. Theor. Probab., Volume 16 (2003) no. 4, pp. 1015-1046 | Article | MR 2033196 | Zbl 1044.60083

[MM07] Marckert, Jean-François; Miermont, Grégory Invariance principles for random bipartite planar maps, Ann. Probab., Volume 35 (2007) no. 5, pp. 1642-1705 | MR 2349571 | Zbl 1208.05135

[MW08] Miermont, Grégory; Weill, Mathilde Radius and profile of random planar maps with faces of arbitrary degrees, Electron. J. Probab., Volume 13 (2008), pp. 79-106 | MR 2375600 | Zbl 1190.60024