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

Ewain Gwynne, Nina Holden, Jason Miller

Research output: Contribution to journalArticlepeer-review


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 κ curve for 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 SLE κ curve for κ∈ (0 , 4) ∪ (4 , 8) and the dimension of the same set with respect to the γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, γ=κ∧(4/κ).

Original languageEnglish (US)
Pages (from-to)649-667
Number of pages19
JournalProbability Theory and Related Fields
Issue number1-2
StatePublished - Feb 1 2020


  • Conformal map
  • Hausdorff dimension
  • KPZ formula
  • Liouville quantum gravity
  • Peanosphere
  • Schramm-Loewner evolution

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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