Simple CLE in doubly connected domains
We restrict attention to CLEκ for which the loops are simple, i.e. κ ∈ (8/3, 4]. In (Ann. of Math. (2) 176 (2012) 1827-1917), simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit di...
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Institute of Mathematical Statistics
2018
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| Online Access: | http://hdl.handle.net/1721.1/116410 https://orcid.org/0000-0002-5951-4933 |
| _version_ | 1811087637645099008 |
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| author | Sheffield, Scott Roger Watson, Samuel Stewart Wu, Hao |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger Watson, Samuel Stewart Wu, Hao |
| author_sort | Sheffield, Scott Roger |
| collection | MIT |
| description | We restrict attention to CLEκ for which the loops are simple, i.e. κ ∈ (8/3, 4]. In (Ann. of Math. (2) 176 (2012) 1827-1917), simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLE's, along with the corresponding exploration processes. |
| first_indexed | 2024-09-23T13:49:38Z |
| format | Article |
| id | mit-1721.1/116410 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T13:49:38Z |
| publishDate | 2018 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1164102022-09-28T16:27:43Z Simple CLE in doubly connected domains Sheffield, Scott Roger Watson, Samuel Stewart Wu, Hao Massachusetts Institute of Technology. Department of Mathematics Sheffield, Scott Roger Watson, Samuel Stewart Wu, Hao Schramm Loewner Evolution, Conformal Loop Ensemble, doubly connected domains, explo- ration process We restrict attention to CLEκ for which the loops are simple, i.e. κ ∈ (8/3, 4]. In (Ann. of Math. (2) 176 (2012) 1827-1917), simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLE's, along with the corresponding exploration processes. 2018-06-19T14:48:44Z 2018-06-19T14:48:44Z 2015-11 2015-10 2018-05-30T15:16:13Z Article http://purl.org/eprint/type/JournalArticle 0246-0203 http://hdl.handle.net/1721.1/116410 Sheffield, Scott, Samuel S. Watson, and Hao Wu. “Simple CLE in Doubly Connected Domains.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 53, no. 2 (May 2017): 594–615. https://orcid.org/0000-0002-5951-4933 http://dx.doi.org/10.1214/15-AIHP726 Annales de l'Institut Henri Poincaré, Probabilités et Statistiques Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Schramm Loewner Evolution, Conformal Loop Ensemble, doubly connected domains, explo- ration process Sheffield, Scott Roger Watson, Samuel Stewart Wu, Hao Simple CLE in doubly connected domains |
| title | Simple CLE in doubly connected domains |
| title_full | Simple CLE in doubly connected domains |
| title_fullStr | Simple CLE in doubly connected domains |
| title_full_unstemmed | Simple CLE in doubly connected domains |
| title_short | Simple CLE in doubly connected domains |
| title_sort | simple cle in doubly connected domains |
| topic | Schramm Loewner Evolution, Conformal Loop Ensemble, doubly connected domains, explo- ration process |
| url | http://hdl.handle.net/1721.1/116410 https://orcid.org/0000-0002-5951-4933 |
| work_keys_str_mv | AT sheffieldscottroger simplecleindoublyconnecteddomains AT watsonsamuelstewart simplecleindoublyconnecteddomains AT wuhao simplecleindoublyconnecteddomains |