Harmonic measure and winding of conformally invariant curves

The exact joint multifractal distribution for the scaling and winding of the electrostatic potential lines near any conformally invariant scaling curve is derived in two dimensions. Its spectrum f(alpha,lambda) gives the Hausdorff dimension of the points where the potential scales with distance r as H approximately r(alpha) while the curve logarithmically spirals with a rotation angle phi=lambdalnr. It obeys the scaling law f(alpha,lambda)=(1+lambda(2))f(alpha)-blambda(2) with alpha=alpha/(1+lambda(2)) and b=(25-c)/12, and where f(alpha) identical with f(alpha,0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to stochastic L?wner evolution.