THE HAUSDORFF DIMENSION OF THE CLE GASKET

• 成果类型: Article
• 署名作者: Miller, Jason; Sun, Nike; Wilson, David B.
• 署名单位: Massachusetts Institute of Technology (MIT); Stanford University; Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP820
• 发表日期: 2014
• 页码: 1644-1665
• 关键词: erased random-walks; conformal-invariance; critical percolation

摘要

The conformal loop ensemble CLEK is the canonical conformally invariant probability measure on noncrossing loops in a proper simply connected domain in the complex plane. The parameter K varies between 8/3 and 8; CLE8/3 is empty while CLE8 is a single space-filling loop. In this work, we study the geometry of the CLE gasket, the set of points not surrounded by any loop of the CLE. We show that the almost sure Hausdorff dimension of the gasket is bounded from below by 2 (8 K)(3K 8)/(32K) when 4 <8. Together with the work of Schramm Sheffield Wilson [Comm. Math. Phys. 288 (2009) 43-53] giving the upper bound for all K and the work of Nacu Werner [J. Lond. Math. Soc. (2) 83 (2011) 789-809] giving the matching lower bound for K <4, this completes the determination of the CLEK gasket dimension for all values of K for which it is defined. The dimension agrees with the prediction of Duplantier Saleur [Phys. Rev. Lett. 63 (1989) 2536 2537] for the FK gasket.