Publication Date:
2019-11-02
Description:
We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $${mathbb {R}}$$R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an $$hbox {SLE}_kappa $$SLEκ curve for $$kappa
ot =4$$κ≠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 $$hbox {SLE}_kappa $$SLEκ curve for $$kappa in (0,4)cup (4,8)$$κ∈(0,4)∪(4,8) and the dimension of the same set with respect to the $$gamma $$γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, $$gamma = sqrt{kappa }wedge (4/sqrt{kappa })$$γ=κ∧(4/κ).
Print ISSN:
0178-8051
Electronic ISSN:
1432-2064
Topics:
Mathematics
Permalink