Quasisymmetric geometry of low-dimensional random spaces

TL;DR

This paper explores the quasisymmetric uniformization of random fractals, demonstrating that many such spaces, including Brownian motion traces and SLE variants, do not conform to simple canonical models.

Contribution

It introduces the first systematic study of quasisymmetric uniformization for natural random fractals, revealing their complex geometric structures.

Findings

Brownian motion trace is almost surely not a quasiarc.

SLE$_ppa$ variants are not quasiarcs for ppa>0.

The outside points of CLE$_ppa$ form a Sierpi
4dnski carpet not quasisymmetrically round.

Abstract

We initiate a study of the quasisymmetric uniformization of naturally arising random fractals and show that many of them fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with the trace, the graph of Brownian motion, and various variants of the Schramm-Loewner evolution $\mathrm{SLE}_\kappa$ for $\kappa>0$, and show that a.s. neither is a quasiarc. After that, we study the conformal loop ensemble $\mathrm{CLE}_\kappa$, $\kappa \in (\frac{8}{3}, 4]$, and show that the collection of all points outside the loops is a.s. homeomorphic to the standard Sierpi\'nski carpet, but not quasisymmetrically equivalent to a round carpet.