Open Access
2018 Natural parametrization of SLE: the Gaussian free field point of view
Stéphane Benoist
Electron. J. Probab. 23: 1-16 (2018). DOI: 10.1214/18-EJP232


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.


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Stéphane Benoist. "Natural parametrization of SLE: the Gaussian free field point of view." Electron. J. Probab. 23 1 - 16, 2018.


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

zbMATH: 06970408
MathSciNet: MR3870446
Digital Object Identifier: 10.1214/18-EJP232

Primary: 60J67 , 60K35

Keywords: Gaussian free field , GFF , LQG , Natural parametrization , SLE

Vol.23 • 2018
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