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: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Institute of Mathematical Statistics
2013
|
| Online Access: | http://hdl.handle.net/1721.1/81178 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1826208302342602752 |
|---|---|
| author | Lawler, Gregory F. Sheffield, Scott Sheffield, Scott Roger |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Lawler, Gregory F. Sheffield, Scott Sheffield, Scott Roger |
| author_sort | Lawler, Gregory F. |
| collection | MIT |
| description | 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... |
| first_indexed | 2024-09-23T14:03:39Z |
| format | Article |
| id | mit-1721.1/81178 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:03:39Z |
| publishDate | 2013 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/811782022-10-01T18:54:13Z A natural parametrization for the Schramm–Loewner evolution Lawler, Gregory F. Sheffield, Scott Sheffield, Scott Roger Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger 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) 2013-09-25T19:08:33Z 2013-09-25T19:08:33Z 2011-09 Article http://purl.org/eprint/type/JournalArticle 0091-1798 http://hdl.handle.net/1721.1/81178 Lawler, Gregory F., and Scott Sheffield. “A natural parametrization for the Schramm–Loewner evolution.” The Annals of Probability 39, no. 5 (September 2011): 1896-1937. https://orcid.org/0000-0002-5951-4933 en_US http://dx.doi.org/10.1214/10-aop560 The Annals of Probability application/pdf Institute of Mathematical Statistics |
| spellingShingle | Lawler, Gregory F. Sheffield, Scott Sheffield, Scott Roger A natural parametrization for the Schramm–Loewner evolution |
| title | A natural parametrization for the Schramm–Loewner evolution |
| title_full | A natural parametrization for the Schramm–Loewner evolution |
| title_fullStr | A natural parametrization for the Schramm–Loewner evolution |
| title_full_unstemmed | A natural parametrization for the Schramm–Loewner evolution |
| title_short | A natural parametrization for the Schramm–Loewner evolution |
| title_sort | natural parametrization for the schramm loewner evolution |
| url | http://hdl.handle.net/1721.1/81178 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT lawlergregoryf anaturalparametrizationfortheschrammloewnerevolution AT sheffieldscott anaturalparametrizationfortheschrammloewnerevolution AT sheffieldscottroger anaturalparametrizationfortheschrammloewnerevolution AT lawlergregoryf naturalparametrizationfortheschrammloewnerevolution AT sheffieldscott naturalparametrizationfortheschrammloewnerevolution AT sheffieldscottroger naturalparametrizationfortheschrammloewnerevolution |