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

2017

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

Benoist, Stéphane

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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.

Built on

The dimension of the SLE curves
Vincent Beffara · 2008
Earlier work this paper cites.
Liouville quantum gravity and KPZ
Bertrand Duplantier and Scott Sheffield · 2011
Earlier work this paper cites.
A natural parametrization for the Schramm-Loewner evolution
Gregory F. Lawler and Scott Sheffield · 2011
Earlier work this paper cites.
Liouville quantum gravity as a mating of trees
Bertrand Duplantier, Jason Miller, and Scott Sheffield · 2014
Earlier work this paper cites.

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