Conformal spiral multifractals

The exact joint multifractal distribution for the scaling and spiraling of 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 similar to r(alpha) while the curve spirals logarithmically 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) equivalent to f(alpha, lambda = 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 SLEk.