Almost sure multifractal spectrum of Schramm–Loewner evolution

Suppose that eta is a Schramm-Loewner evolution (SLE kappa) in a smoothly bounded simply connected domain D subset of C and that phi is a conformal map from D to a connected component of D\eta([0, t]) for some t > 0. The multifractal spectrum of eta is the function. (-1, 1) -> [0, infinity) which, for each s is an element of(-1, 1) gives the Hausdorff dimension of the set of points x is an element of partial derivative D such that vertical bar phi '((1 - is an element of)X)vertical bar = is an element of(-s+o(1)) as is an element of -> 0. We rigorously compute the almost sure multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of SLE, we obtain the optimal Holder exponent for a conformal map which uniformizes the complement of an SLE curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the SLE curve for kappa <= 4. Our results also hold for the SLE kappa(rho) processes with general vectors of weight rho.