Convergence of Eulerian triangulations

TL;DR

This paper proves that large planar Eulerian triangulations, when properly rescaled, converge to the Brownian map, using adapted layer decomposition methods to handle their unique pseudo-distance structure.

Contribution

It introduces a novel approach to prove convergence of Eulerian triangulations to the Brownian map, overcoming challenges posed by their oriented pseudo-distance.

Findings

Convergence of Eulerian triangulations to the Brownian map established

New models of infinite random maps constructed as local limits

First proof of convergence with Riemannian metric for such maps

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 established by Curien and Le Gall, 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.