A natural parametrization for the Schramm–Loewner evolution
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Sheffield_A natural parametrization.pdf
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Author(s)
Lawler, Gregory F.
Sheffield, Scott
Sheffield, Scott Roger
Date Issued
September 2011
Journal
The Annals of Probability
Publisher
Institute of Mathematical Statistics
Citation
Lawler, Gregory F., and Scott Sheffield. “A natural parametrization for the Schramm–Loewner evolution.” The Annals of Probability 39, no. 5 (September 2011): 1896-1937.
Version
Author's final manuscript
Abstract
The Schramm–Loewner evolution (SLE[subscript κ]) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ < 8, an instance of SLE[subscript κ] is a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume.
For κ<8, we use a Doob–Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE[subscript κ] that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is nontrivial (i.e., the curve is not entirely traversed in zero time) for k <4(7-√33)=5.021...
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1214/10-aop560
