The Annals of Probability

Stability of geodesics in the Brownian map

Omer Angel, Brett Kolesnik, and Grégory Miermont

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Abstract

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.

Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 3451-3479.

Dates
Received: February 2015
Revised: August 2016
First available in Project Euclid: 23 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1506132042

Digital Object Identifier
doi:10.1214/16-AOP1140

Mathematical Reviews number (MathSciNet)
MR3706747

Zentralblatt MATH identifier
06812209

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 05C80: Random graphs [See also 60B20] 54E52: Baire category, Baire spaces 54G99: None of the above, but in this section

Keywords
Brownian map planar map geodesic cut locus real tree scaling limit Gromov–Hausdorff convergence random geometry quantum gravity

Citation

Angel, Omer; Kolesnik, Brett; Miermont, Grégory. Stability of geodesics in the Brownian map. Ann. Probab. 45 (2017), no. 5, 3451--3479. doi:10.1214/16-AOP1140. https://projecteuclid.org/euclid.aop/1506132042


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