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2021 Convergence of Eulerian triangulations
Ariane Carrance
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Electron. J. Probab. 26: 1-48 (2021). DOI: 10.1214/21-EJP579


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.


I warmly thank Grégory Miermont for his crucial help and advice, as well as Nicolas Curien and Christina Goldschmidt for their insightful remarks. I also thank the referees for their very helpful feedback.


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Ariane Carrance. "Convergence of Eulerian triangulations." Electron. J. Probab. 26 1 - 48, 2021.


Received: 14 January 2020; Accepted: 10 January 2021; Published: 2021
First available in Project Euclid: 23 March 2021

arXiv: 1912.13434
Digital Object Identifier: 10.1214/21-EJP579

Primary: 05A16 , 05C80 , 60B05 , 60J80

Keywords: branching processes , local limits of maps , Random maps , scaling limits of maps


Vol.26 • 2021
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