An axiomatic characterization of the Brownian map

TL;DR

This paper provides an axiomatic characterization of the Brownian map, a fundamental object in random geometry, using a new breadth-first construction and properties like scale invariance and slice independence.

Contribution

It introduces an alternative breadth-first construction of the Brownian map and characterizes it uniquely through axioms involving scale invariance and conditional independence.

Findings

Proves the Brownian map is the unique sphere-homeomorphic metric measure space with specified properties.

Connects the Brownian map to the Le9vy net and metric exploration processes.

Lays groundwork for proving equivalence with Liouville quantum gravity.

Abstract

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. We also formulate a characterization in terms of the so-called L\'evy net produced by a metric exploration from one measure-typical point to another. This characterization is part of a program for proving the equivalence of the Brownian map and Liouville quantum gravity with parameter $\gamma= \sqrt{8/3}$.