Stability of geodesics in the Brownian map

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.

arXiv comment29 pages, 7 figures, final version

References in corpus (12)

• Uniqueness and universality of the Brownian map
• The topological structure of scaling limits of large planar maps
• Uniform Infinite Planar Triangulations
• Local structure of random quadrangulations
• Scaling limits of random trees and planar maps
• The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection
• A view from infinity of the uniform infinite planar quadrangulation
• Quadrangulations with no pendant vertices
• The topology of scaling limits of positive genus random quadrangulations
• The scaling limit of random simple triangulations and random simple quadrangulations
• Geodesics in Brownian surfaces (Brownian maps)
• On one property of distances in the infinite random quadrangulation

Cited by in corpus (5)

• Geodesic networks in Liouville quantum gravity surfaces
• Geodesics in the Brownian map: Strong confluence and geometric structure
• Uniqueness of the critical and supercritical Liouville quantum gravity metrics
• Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric
• Introduction to the Liouville quantum gravity metric