The Annals of Probability

A natural parametrization for the Schramm–Loewner evolution

Gregory F. Lawler and Scott Sheffield

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Abstract

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 $\kappa\textless 4(7-\sqrt{33})=5.021\ldots$.

Article information

Source
Ann. Probab. Volume 39, Number 5 (2011), 1896-1937.

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
http://projecteuclid.org/euclid.aop/1318940785

Digital Object Identifier
doi:10.1214/10-AOP560

Zentralblatt MATH identifier
05987673

Mathematical Reviews number (MathSciNet)
MR2884877

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

Keywords
Schramm–Loewner evolution natural parametrization

Citation

Lawler, Gregory F.; Sheffield, Scott. A natural parametrization for the Schramm–Loewner evolution. The Annals of Probability 39 (2011), no. 5, 1896--1937. doi:10.1214/10-AOP560. http://projecteuclid.org/euclid.aop/1318940785.


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