Electron. J. Probab. 26 (2021), article no. 18, 1–48. ISSN: 1083-6489 https://doi.org/10.1214/21-EJP579
Convergence of Eulerian triangulations
Ariane Carrance*
Abstract
We prove that properly rescaled large planar Eulerian triangulations converge to the Brownian map. This result requires more than a standard application of the methods that have been used to obtain the convergence of other families of planar maps to the Brownian map, as the natural distance for Eulerian triangulations is a canonical oriented pseudo-distance. To circumvent this difficulty, we adapt the layer decomposition method, as formalized by Curien and Le Gall in [13], which yields asymptotic proportionality between three natural distances on planar Eulerian triangulations: the usual graph distance, the canonical oriented pseudo-distance, and the Riemannian metric. This notably gives the first mathematical proof of a convergence to the Brownian map for maps endowed with their Riemannian metric. Along the way, we also construct new models of infinite random maps, as local limits of large planar Eulerian triangulations.
Keywords: random maps; scaling limits of maps; local limits of maps; branching processes. MSC2020 subject classifications: 05C80; 60B05; 60J80; 05A16. Submitted to EJP on January 14, 2020, final version accepted on January 10, 2021. Supersedes arXiv:1912.13434.
1 Introduction
1.1 Context
Eulerian triangulations are face-bicolored triangulations. They can be encountered in several contexts. As their definition is quite straightforward, they are already an object of interest in themselves in enumerative combinatorics (see [27, 10, 8, 4]). Moreover, they are in bijection with combinatorial objects such as constellations and bipartite maps, and geometrical objects such as Belyi surfaces (see [19]). They also correspond to the two-dimensional case of colored tensor models, an approach to quantum gravity that generalizes matrix models to any dimension (see Part I of [12] for an introduction to this topic).
The main aim of this paper is to show that large planar rooted Eulerian triangulations converge to the Brownian map (see Theorem 3.1 for a more precise statement). Along the
*Université Paris-Saclay, France. Supported by the ERC Advanced Grant 740943 GeoBrown. E-mail: ariane.carrance@math.cnrs.fr