Co-authors: Jason Miller (Massachusetts Institute of Technology), Xin Sun (Massachusetts Institute of Technology)
Suppose that η is an SLEκ in a smoothly bounded simply connected domain D⊂C and that ϕ is a conformal map from the unit disk D to a connected component of D∖η([0,t]) for some t>0. The multifractal spectrum of η is the function (−1,1)→[0,∞) which, for each s∈(−1,1), gives the Hausdorff dimension of the set of points x∈∂D such that |ϕ′((1−ϵ)x)|=ϵ−s+o(1) as ϵ→0. I will present a rigorous computation of the a.s. multifractal spectrum of SLE (joint with J. Miller and X. Sun), which confirms a prediction due to Duplantier. The proof makes use of various couplings of SLE with the Gaussian free field. As a corollary, we also confirm a conjecture of Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE.
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