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2010 Geodesics in large planar maps and in the Brownian map
Jean-François Gall
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Acta Math. 205(2): 287-360 (2010). DOI: 10.1007/s11511-010-0056-5


We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.


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Jean-François Gall. "Geodesics in large planar maps and in the Brownian map." Acta Math. 205 (2) 287 - 360, 2010.


Received: 2 June 2008; Revised: 26 June 2009; Published: 2010
First available in Project Euclid: 31 January 2017

zbMATH: 1214.53036
MathSciNet: MR2746349
Digital Object Identifier: 10.1007/s11511-010-0056-5

Rights: 2010 © Institut Mittag-Leffler

Vol.205 • No. 2 • 2010
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