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September 2017 Stability of geodesics in the Brownian map
Omer Angel, Brett Kolesnik, Grégory Miermont
Ann. Probab. 45(5): 3451-3479 (September 2017). DOI: 10.1214/16-AOP1140


The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure.

Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a closed, nowhere dense set of zero measure.

We obtain similar stability results for the set of points inside geodesics to a fixed point. Furthermore, we show that the set of points inside geodesics of the map is of first Baire category. Hence, most points in the Brownian map are endpoints.

Finally, we classify the types of geodesic networks which are dense. For each $k\in\{1,2,3,4,6,9\}$, there is a dense set of pairs of points which are joined by networks of exactly $k$ geodesics and of a specific topological form. We find the Hausdorff dimension of the set of pairs joined by each type of network. All other geodesic networks are nowhere dense.


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Omer Angel. Brett Kolesnik. Grégory Miermont. "Stability of geodesics in the Brownian map." Ann. Probab. 45 (5) 3451 - 3479, September 2017.


Received: 1 February 2015; Revised: 1 August 2016; Published: September 2017
First available in Project Euclid: 23 September 2017

zbMATH: 06812209
MathSciNet: MR3706747
Digital Object Identifier: 10.1214/16-AOP1140

Primary: 60D05
Secondary: 05C80 , 54E52 , 54G99

Keywords: Brownian map , Cut locus , Geodesic , Gromov–Hausdorff convergence , Planar map , quantum gravity , Random geometry , real tree , Scaling limit

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • September 2017
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