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• W1822608318 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 nowhere dense set. 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 $kin{1,2,3,4,6,9}$, 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." @default.
• W1822608318 created "2016-06-24" @default.
• W1822608318 creator A5001338460 @default.
• W1822608318 creator A5061258160 @default.
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• W1822608318 date "2015-02-16" @default.
• W1822608318 modified "2023-09-27" @default.
• W1822608318 title "On the stability of geodesics in the Brownian map" @default.
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• W1822608318 hasPublicationYear "2015" @default.
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