Stability of geodesics in the Brownian map

TL;DR

This paper investigates the stability and structure of geodesics in the Brownian map, revealing continuity, stability, and classification of geodesic networks, and showing most points are endpoints with complex dense geodesic networks.

Contribution

It establishes the continuity and stability of the cut locus in the Brownian map and classifies dense geodesic networks, advancing understanding of its geodesic structure.

Findings

The cut locus is continuous at typical points.

The cut locus is stable outside a measure-zero nowhere dense set.

Most points are endpoints, with dense geodesic networks of specific types.

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.