The Brownian map and Brownian continuum random tree are important random metric spaces that have seen a lot of interest in recent years. The Brownian map in particular arises as the limiting object of rescaled random quadrangulations of the sphere with potential links to quantum gravity. The Brownian map is topologically a sphere and we are motivated by the question of finding a canonical embedding from the abstract random metric space to a simpler, say Euclidean, space.

While the Brownian map arises as the scaling limit of planar maps, the Brownian continuum random tree (BCRT) is the rescaled limit of critical Galton-Watson trees. While at first glance there may not be a link between the two, the Brownian map can be obtained from the BCRT by a change in metric.

In this talk I will introduce the two objects, describe their relationship and give a dimension theorist’s perspective to these random metric spaces. In particular, I will show that there cannot exist a quasi-symmetric embedding of the Brownian map / BCRT into R^d for any d.