Natural parametrization of SLE: the Gaussian free field point of view

TL;DR

This paper introduces a new construction of the natural parametrization of SLE$_ppa$ for ppa<4 using Gaussian free fields, linking quantum time and Liouville boundary measures to characterize the process.

Contribution

It presents a novel construction of SLE natural parametrization via quantum time derived from Gaussian free fields, offering a new proof of its Markovian covariance property.

Findings

Quantum time equals expectation of the quantum measure on SLE

Quantum time can be reconstructed as a chaos with Markovian covariance

Provides an alternative proof of the characterization of natural parametrization

Abstract

We provide another construction of the natural parametrization of SLE$_\kappa$ for $\kappa < 4$. We construct it as the expectation of the quantum time, which is a random measure carried by SLE in an ambient Gaussian free field. This quantum time was built as the push forward on the SLE curve of the Liouville boundary measure, which is a natural field-dependent measure supported on the boundary of the domain. We moreover show that the quantum time can be reconstructed as a chaos on any measure on the trace of SLE with the right Markovian covariance property. This provides another proof that natural parametrization is characterized by its Markovian covariance property.