On quasisymmetric embeddings of the Brownian map and continuum trees

TL;DR

This paper proves that the Brownian map and continuum random trees cannot be embedded into common metric spaces via quasisymmetric maps, using dimension theory to identify invariant structures with infinite Assouad dimension.

Contribution

It demonstrates the impossibility of quasisymmetric embeddings of the Brownian map and continuum trees into standard metric spaces, revealing intrinsic fractal properties.

Findings

Any embedding into Euclidean or doubling spaces cannot be quasisymmetric.

The Brownian map and continuum trees almost surely have infinite Assouad dimension.

Snowflaking does not enable quasisymmetric embedding of these structures.

Abstract

The Brownian map is a model of random geometry on the sphere and as such an important object in probability theory and physics. It has been linked to Liouville Quantum Gravity and much research has been devoted to it. One open question asks for a canonical embedding of the Brownian map into the sphere or other, more abstract, metric spaces. Similarly, Liouville Quantum Gravity has been shown to be "equivalent" to the Brownian map but the exact nature of the correspondence (i.e.\ embedding) is still unknown. In this article we show that any embedding of the Brownian map or continuum random tree into $\mathbb{R}^d$, $\mathbb{S}^d$, $\mathbb{T}^d$, or more generally any doubling metric space, cannot be quasisymmetric. We achieve this with the aid of dimension theory by identifying a metric structure that is invariant under quasisymmetric mappings (such as isometries) and which implies infinite Assouad dimension. We show, using elementary methods, that this structure is almost surely present in the Brownian continuum random tree and the Brownian map. We further show that snowflaking the metric is not sufficient to find an embedding and discuss continuum trees as a tool to studying "fractal functions".