An axiomatic characterization of the Brownian map

Understand

The Brownian map is a random sphere-homeomorphic metric measure space obtained by "gluing together" the continuum trees described by the $x$ and $y$ coordinates of the Brownian snake.

• We present an alternative "breadth-first" construction of the Brownian map, which produces a surface from a certain decorated branching process.
• It is closely related to the peeling process, the hull process, and the Brownian cactus.
• Using these ideas, we prove that the Brownian map is the only random sphere-homeomorphic metric measure space with certain properties: namely, scale invariance and the conditional independence of the inside and outside of certain "slices" bounded by geodesics.