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 {{,mathrm{mathbb {R}},}}^d, {{,mathrm{mathbb {S}},}}^d, {{,mathrm{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”.
Highlights
Over the past few years two important models of random geometry of the sphere S2 emerged
The Brownian map turned out to be a universal limit of random planar maps of S2 and both models have attracted a great deal of interest over the past few years
The Brownian map is homeomorphic to S2 but has Hausdorff dimension 4, indicating that the homeomorphism is highly singular. Finding such a canonical mapping is still an active research area and in this article we show, using elementary facts from probability and dimension theory, that the Brownian map has an almost sure metric property that we coin starry. This property implies that the Brownian map and its images under quasisymmetric mappings has infinite Assouad dimension and cannot be embedded by quasisymmetric mappings into finite dimensional manifolds
Summary
Over the past few years two important models of random geometry of the sphere S2 emerged. Finding such a canonical mapping is still an active research area and in this article we show, using elementary facts from probability and dimension theory, that the Brownian map has an almost sure metric property that we coin starry This property implies that the Brownian map and its images under quasisymmetric mappings has infinite Assouad dimension and cannot be embedded by quasisymmetric mappings (such as isometries) into finite dimensional manifolds. 3, we define the Brownian continuum random tree via the Brownian excursion and show that the CRT is starry almost surely We conclude that it cannot be embedded into Rd for any d ∈ N using quasisymmetric mappings. Theorem 2.4 states that quasisymmetric images of starry metric spaces have infinite Assouad dimension (and are not doubling), Theorem 3.5 proves that the CRT is starry almost surely, and Theorem 4.1 shows that the Brownian map is starry. Throughout, we postpone proofs until the end of their respective section
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