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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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Institute of Mathematical Statistics
2013
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| Online Access: | http://hdl.handle.net/1721.1/81178 https://orcid.org/0000-0002-5951-4933 |
| 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... |
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