Acta Mathematica

Geodesics in large planar maps and in the Brownian map

Jean-François Gall

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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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Acta Math., Volume 205, Number 2 (2010), 287-360.

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

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2010 © Institut Mittag-Leffler


Gall, Jean-François. Geodesics in large planar maps and in the Brownian map. Acta Math. 205 (2010), no. 2, 287--360. doi:10.1007/s11511-010-0056-5.

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