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Continuity and stability of the cut locus of the Brownian map

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A prototype for pure quantum gravity is the Brownian map, a random geodesic metric space which is homeomorphic to the sphere, of Hausdorff dimension 4, and the scaling limit of a wide variety of 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 on an open, dense subset of the Brownian map. Moreover, the cut locus is uniformly stable in the sense that any two cut loci coincide outside a nowhere dense set. Other consequences include the classification of geodesic networks which are dense. For each j,k in {1,2,3}, there is a dense set of Hausdorff dimension 2(6-j-k) of pairs of points which are joined by networks of exactly jk geodesics and of a specific topological form. All other networks are nowhere dense.

Joint work with Omer Angel (UBC) and Gregory Miermont (ENS Lyon and IUF )

This talk is part of the Probability series.

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