The conformal dimension of the Brownian tree is one

TL;DR

This paper proves that the conformal dimension of the Brownian tree, a fundamental random metric space, is exactly 1, aligning with its topological dimension.

Contribution

It establishes that the conformal dimension of the Brownian tree is equal to 1, a significant invariant in geometric analysis of random trees.

Findings

Conformal dimension of the Brownian tree is 1.

Matches the topological dimension of the space.

Provides insight into the geometric structure of the Brownian tree.

Abstract

The Brownian tree, also known as the continuum random tree, is a canonical random compact, geodesic $\mathbf R$-tree that arises as the universal scaling limit for numerous models of discrete random trees. A key quasisymmetric invariant of a metric space is its conformal dimension, defined as the infimum of the Hausdorff dimensions over all quasisymmetrically equivalent spaces. This value is always bounded below by the space's topological dimension and above by its Hausdorff dimension. In the present paper, we prove that the conformal dimension of the Brownian tree is $1$, matching its topological dimension.