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 $${{\,\mathrm{\mathbb {R}}\,}}^d$$ R d , $${{\,\mathrm{\mathbb {S}}\,}}^d$$ S d , $${{\,\mathrm{\mathbb {T}}\,}}^d$$ 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”.

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关于布朗图和连续树的准对称嵌入

布朗图是球体上的随机几何模型，因此是概率论和物理学中的重要对象。它与刘维尔量子引力有关，并且有很多研究致力于它。一个开放性问题要求将布朗映射规范嵌入到球体或其他更抽象的度量空间中。类似地，Liouville Quantum Gravity 已被证明与布朗图“等效”，但对应的确切性质（即嵌入）仍然未知。在本文中，我们展示了将布朗图或连续随机树嵌入到 $${{\,\mathrm{\mathbb {R}}\,}}^d$$ R d , $${{\,\ mathrm{\mathbb {S}}\,}}^d$$ S d , $${{\,\mathrm{\mathbb {T}}\,}}^d$$ T d ，或更一般的任何加倍度量空间，不能是拟对称的。我们在维度理论的帮助下实现了这一点，通过识别在准对称映射（例如等距）下不变的度量结构，这意味着无限的 Assouad 维度。我们使用基本方法表明，这种结构几乎肯定存在于布朗连续随机树和布朗地图中。我们进一步表明，雪花度量不足以找到嵌入并讨论连续树作为研究“分形函数”的工具。