Optimal Hölder exponent for the SLE path

Abstract

We prove an upper bound on the optimal Hölder exponent for the chordal $\mathrm{SLE}$ path parameterized by capacity and thereby establish the optimal exponent as conjectured by Lind. We also give a new proof of the lower bound. Our proofs are based on sharp estimates of moments of the derivative of the inverse map. In particular, we improve an estimate of the second author.