An axiomatic characterization of the Brownian map
Jason Miller, Scott Sheffield⋆
Abstract
The Brownian map is a random sphere-homeomorphic metric measure space obtained by “gluing together” the continuum trees described by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>y</mml:mi> </mml:math> 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 and metric ball boundaries. We also formulate a characterization in terms of the so-called Lévy 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 the Liouville quantum gravity sphere with parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>8</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> .