The natural parametrization for the Schramm-Loewner evolution
Abstract
The Schramm-Loewner evolution (SLE_\kappa) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When \kappa < 8, an instance of SLE_\kappa is a random planar curve with almost sure Hausdorff dimension d = 1 + \kappa/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For \kappa < 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_\kappa that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is non-trivial (i.e., the curve is not entirely traversed in zero time) for \kappa < 4(7 - \sqrt{33}) = 5.021 ....
Cite
@article{arxiv.0906.3804,
title = {The natural parametrization for the Schramm-Loewner evolution},
author = {Gregory F. Lawler and Scott Sheffield},
journal= {arXiv preprint arXiv:0906.3804},
year = {2009}
}