Conformal Spiral Multifractals

Abstract

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(α,λ) gives the Hausdorff dimension of the points where the potential scales with distance r as$H \sim {r^\alpha }$while the curve spirals logarithmically with a rotation angleφ=λln r. It obeys the scaling law $f\left( {\alpha ,\lambda } \right) = \left( {1 + {{\lambda }^{2}}} \right)f\left( {\bar{\alpha }} \right) - b{{\lambda }^{2}}$ with $\bar{\alpha } = \alpha /\left( {1 + {{\lambda }^{2}}} \right)$ and b=(25-c)/12, and where f(α) Ξ f(α, λ=0) is the pure harmonic measure spectrum, andcthe conformal central charge. The results apply toO(N)and Potts models, as well as to SLE,,.