The conformal loop ensemble nesting field
The conformal loop ensemble CLE[subscript κ]with parameter 8/3<κ<8 is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops surrounding an ε-ball (a random function of z and ε) minus it...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer-Verlag
2016
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| Online Access: | http://hdl.handle.net/1721.1/104898 |
| _version_ | 1826206663539949568 |
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| author | Wilson, David B. Miller, Jason P. Watson, Samuel Stewart |
| author2 | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory |
| author_facet | Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Wilson, David B. Miller, Jason P. Watson, Samuel Stewart |
| author_sort | Wilson, David B. |
| collection | MIT |
| description | The conformal loop ensemble CLE[subscript κ]with parameter 8/3<κ<8 is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops surrounding an ε-ball (a random function of z and ε) minus its expectation converges almost surely as ε→0 to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points. |
| first_indexed | 2024-09-23T13:36:19Z |
| format | Article |
| id | mit-1721.1/104898 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:36:19Z |
| publishDate | 2016 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/1048982022-10-01T16:00:45Z The conformal loop ensemble nesting field Wilson, David B. Miller, Jason P. Watson, Samuel Stewart Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology. Department of Mathematics Miller, Jason P. Watson, Samuel Stewart The conformal loop ensemble CLE[subscript κ]with parameter 8/3<κ<8 is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops surrounding an ε-ball (a random function of z and ε) minus its expectation converges almost surely as ε→0 to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points. 2016-10-20T20:27:09Z 2016-10-20T20:27:09Z 2015-03 2014-11 2016-08-18T15:27:46Z Article http://purl.org/eprint/type/JournalArticle 0178-8051 1432-2064 http://hdl.handle.net/1721.1/104898 Miller, Jason and Samuel S. Watson, and David B. Wilson."The conformal loop ensemble nesting field." Probability Theory and Related Fields, vol. 163, no. 3, March 2015, pp. 769-801. en http://dx.doi.org/10.1007/s00440-014-0604-6 Probability Theory and Related Fields Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer-Verlag Berlin Heidelberg application/pdf Springer-Verlag Springer Berlin Heidelberg |
| spellingShingle | Wilson, David B. Miller, Jason P. Watson, Samuel Stewart The conformal loop ensemble nesting field |
| title | The conformal loop ensemble nesting field |
| title_full | The conformal loop ensemble nesting field |
| title_fullStr | The conformal loop ensemble nesting field |
| title_full_unstemmed | The conformal loop ensemble nesting field |
| title_short | The conformal loop ensemble nesting field |
| title_sort | conformal loop ensemble nesting field |
| url | http://hdl.handle.net/1721.1/104898 |
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