Geodesics in large planar maps and in the Brownian map

Jean-François Le Gall Université Paris-Sud, Département de Mathématiques, Bâtiment 425, Orsay Cedex, France

TBD mathscidoc:1701.332021

Acta Mathematica, 205, (2), 287-360, 2008.6
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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  title={Geodesics in large planar maps and in the Brownian map},
  author={Jean-François Le Gall},
  booktitle={Acta Mathematica},
Jean-François Le Gall. Geodesics in large planar maps and in the Brownian map. 2008. Vol. 205. In Acta Mathematica. pp.287-360.
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