Dimension transformation formula for conformal maps into the complement of an SLE curve

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Holden, Nina; Miller, Jason
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00952-y
• 发表日期: 2020
• 页码: 649-667
• 关键词: GAUSSIAN MULTIPLICATIVE CHAOS; sure multifractal spectrum; erased random-walks; harmonic measure; Duality; reversibility; CONVERGENCE; Invariance

摘要

We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an SLE kappa curve for kappa not equal 4. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes and Vargas (ESAIM Probab Stat 15:358-371, 2011) and the KPZ formula of Gwynne et al. (Ann Probab, 2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an SLE kappa curve for kappa is an element of (0, 4) boolean OR (4, 8) and the dimension of the same set with respect to the gamma-quantum natural parameterization of the curve induced by an independent Gaussian free field, gamma = root kappa boolean AND (4/root kappa).