Hausdorff dimensions for SLE6

2002/04/30 by Vincent Beffara · 4 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.CV #math.PR #msc:28A80. #msc:60D05 #msc:60G17

paper · pdf · doi:10.1214/009117904000000072

published as Annals of Probability 2004, Vol. 32, No. 3, 2606-2629 · Published at http://dx.doi.org/10.1214/009117904000000072 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2004/07/01 · arxiv created 2005/03/30 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We prove that the Hausdorff dimension of the trace of SLE6 is almost surely 7/4 and give a more direct derivation of the result (due to Lawler–Schramm–Werner) that the dimension of its boundary is 4/3. We also prove that, for all κ<8, the SLEκ trace has cut-points.

Cited by

• Three-point functions in critical loop models
• SLE(κ,ρ) martingales and duality
• Geometric exponents, SLE and logarithmic minimal models
• Stochastic geometry of critical curves, Schramm–Loewner evolutions and conformal field theory

Related

• Intersections of SLE Paths: the double and cut point dimension of SLE
• Dimension and natural parametrization for SLE curves
• Dimension of the SLE light cone, the SLE fan, and SLE_κ(ρ) for κ∈ (0,4) and ρ∈ [\tfracκ2-4,-2)
• Hausdorff measure of SLE curves
• Continuity of the SLE trace in simply connected domains