The Annals of Probability

A natural parametrization for the Schramm–Loewner evolution

Gregory F. Lawler and Scott Sheffield

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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

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

First available in Project Euclid: 18 October 2011

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Zentralblatt MATH identifier

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

Schramm–Loewner evolution natural parametrization


Lawler, Gregory F.; Sheffield, Scott. A natural parametrization for the Schramm–Loewner evolution. Ann. Probab. 39 (2011), no. 5, 1896--1937. doi:10.1214/10-AOP560.

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