Geodesics in Random Metric Spaces

Description

The Brownian map is the canonical model for a metric space chosen “uniformly at random” among metric spaces which have the topology of the two-dimensional sphere S2. It was proved by Le Gall and Miermont to be the scaling limit of uniformly random quadrangulations, and is believed to be the universal scaling limit of a wide class of planar maps that are chosen uniformly at random. Miller and Sheffield further showed that the Brownian map is equivalent as a metric measure space to Liouville quantum gravity (LQG) with parameter γ = √ 8/3.The current project aims to acquire a fine understanding of the behavior of geodesics in Brownian surfaces, and eventually LQG surfaces. In our previous work with Miller, we proved a strong and quantitative version of confluence of geodesics which applies simultaneously to all geodesics in the Brownian map. We further deduced many basic geometric properties of geodesics. For example, we showed that there can be at most 9 geodesics between any pairs of points. We also reduced all possible configurations of geodesics between any two points (also calledgeodesic networks) to a finite number of possibilities, and gave a dimension upper bound for each configuration. Our first goal is to determine the exact collection of possible geodesic networks in the Brownian map, and compute the exact dimension for each network. More concretely, we need to establish the missing dimension lower bounds from our previous work, and prove the existence of some networks with dimension 0. Our second goal is to study Brownian surfaces with a boundary, and classify all geodesics to the boundary of the surface. Finally, we would like to extend some of our previous results for Brownian surfaces to LQG surfaces.