## Electronic Journal of Probability

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

Stéphane Benoist

#### Abstract

We provide another construction of the natural parametrization of SLE$_\kappa$ [9, 8] for $\kappa < 4$. We construct it as the expectation of the quantum time [14], 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 the natural parametrization is characterized by its Markovian covariance property.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 103, 16 pp.

Dates
Received: 25 August 2017
Accepted: 4 October 2018
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1539828067

Digital Object Identifier
doi:10.1214/18-EJP232

Zentralblatt MATH identifier
06970408

#### Citation

Benoist, Stéphane. Natural parametrization of SLE: the Gaussian free field point of view. Electron. J. Probab. 23 (2018), paper no. 103, 16 pp. doi:10.1214/18-EJP232. https://projecteuclid.org/euclid.ejp/1539828067

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