Geometric Interpretation of Half-Plane Capacity
Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure...
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| Format: | Article |
| Language: | en_US |
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Institute of Mathematical Statistics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89524 |
| _version_ | 1811095739136212992 |
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| author | Lalley, Steven P. Lawler, Gregory F. Narayanan, Hariharan |
| author2 | Massachusetts Institute of Technology. Laboratory for Information and Decision Systems |
| author_facet | Massachusetts Institute of Technology. Laboratory for Information and Decision Systems Lalley, Steven P. Lawler, Gregory F. Narayanan, Hariharan |
| author_sort | Lalley, Steven P. |
| collection | MIT |
| description | Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure of the size of a set in the reference upper half-plane. This has useful harmonic and complex analytic properties and makes SLE a time-homogeneous Markov process on conformal maps. In this note, we show that the half-plane capacity of a hull A is comparable up to multiplicative constants to more geometric quantities, namely the area of the union of all balls centered in A tangent to R, and the (Euclidean) area of a 1-neighborhood of A with respect to the hyperbolic metric. |
| first_indexed | 2024-09-23T16:26:22Z |
| format | Article |
| id | mit-1721.1/89524 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:26:22Z |
| publishDate | 2014 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/895242022-10-02T07:59:25Z Geometric Interpretation of Half-Plane Capacity Lalley, Steven P. Lawler, Gregory F. Narayanan, Hariharan Massachusetts Institute of Technology. Laboratory for Information and Decision Systems Narayanan, Hariharan Schramm-Loewner Evolution describes the scaling limits of interfaces in certain statistical mechanical systems. These interfaces are geometric objects that are not equipped with a canonical parametrization. The standard parametrization of SLE is via half-plane capacity, which is a conformal measure of the size of a set in the reference upper half-plane. This has useful harmonic and complex analytic properties and makes SLE a time-homogeneous Markov process on conformal maps. In this note, we show that the half-plane capacity of a hull A is comparable up to multiplicative constants to more geometric quantities, namely the area of the union of all balls centered in A tangent to R, and the (Euclidean) area of a 1-neighborhood of A with respect to the hyperbolic metric. National Science Foundation (U.S.) (Grant DMS-0805755) National Science Foundation (U.S.) (Grant DMS-0734151) 2014-09-15T15:17:29Z 2014-09-15T15:17:29Z 2009-12 2009-08 Article http://purl.org/eprint/type/JournalArticle 1083-589X http://hdl.handle.net/1721.1/89524 Lalley, Steven P., Gregory F. Lawler, and Hariharan Narayanan. “Geometric Interpretation of Half-Plane Capacity.” Electronic Communications in Probability 14, no. 0 (January 1, 2009). en_US http://dx.doi.org/10.1214/ECP.v14-1517 Electronic Communications in Probability Creative Commons Attribution http://creativecommons.org/licenses/by/2.5/ application/pdf Institute of Mathematical Statistics Institute of Mathematical Statistics |
| spellingShingle | Lalley, Steven P. Lawler, Gregory F. Narayanan, Hariharan Geometric Interpretation of Half-Plane Capacity |
| title | Geometric Interpretation of Half-Plane Capacity |
| title_full | Geometric Interpretation of Half-Plane Capacity |
| title_fullStr | Geometric Interpretation of Half-Plane Capacity |
| title_full_unstemmed | Geometric Interpretation of Half-Plane Capacity |
| title_short | Geometric Interpretation of Half-Plane Capacity |
| title_sort | geometric interpretation of half plane capacity |
| url | http://hdl.handle.net/1721.1/89524 |
| work_keys_str_mv | AT lalleystevenp geometricinterpretationofhalfplanecapacity AT lawlergregoryf geometricinterpretationofhalfplanecapacity AT narayananhariharan geometricinterpretationofhalfplanecapacity |