A natural parametrization for the Schramm-Loewner evolution
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 convent...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematical Statistics,
2013-09-25T19:08:33Z.
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| Subjects: | |
| Online Access: | Get fulltext |
| Summary: | 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... National Science Foundation (U.S.) (Grant DMS-06-45585) National Science Foundation (U.S.) (Grant DMS-04-03182) National Science Foundation (U.S.) (Grant OISE-0730136) |
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