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September 2011 A natural parametrization for the Schramm–Loewner evolution
Gregory F. Lawler, Scott Sheffield
Ann. Probab. 39(5): 1896-1937 (September 2011). DOI: 10.1214/10-AOP560


The Schramm–Loewner evolution (SLEκ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ < 8, an instance of SLEκ 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 SLEκ 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 $\kappa< 4(7-\sqrt{33})=5.021\ldots$.


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Gregory F. Lawler. Scott Sheffield. "A natural parametrization for the Schramm–Loewner evolution." Ann. Probab. 39 (5) 1896 - 1937, September 2011.


Published: September 2011
First available in Project Euclid: 18 October 2011

zbMATH: 1234.60087
MathSciNet: MR2884877
Digital Object Identifier: 10.1214/10-AOP560

Primary: 60J67

Keywords: Natural parametrization , Schramm–Loewner evolution

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 5 • September 2011
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