Fix an odd integer . Let be a uniform -angulation with vertices, endowed with the uniform probability measure on its vertices. We prove that there exists such that, after rescaling distances by , 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.
[Ald91] 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
[Bil13] Probability and measure. Anniversary edition, John Wiley & Sons, 2013 | Zbl 1236.60001
[BJM14] The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection, Electron. J. Probab., Volume 19 (2014), 74 | Zbl 1320.60088
[GPW09] Convergence in distribution of random metric measure spaces (-coalescent measure trees), Probab. Theory Relat. Fields, Volume 145 (2009) no. 1-2, pp. 285-322 | Article | MR 2520129 | Zbl 1215.05161
[Mie06] 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