Harmonic Measure and Winding of Conformally Invariant Curves

TL;DR

This paper derives an exact multifractal distribution for the scaling and winding of electrostatic potential lines near conformally invariant curves in two dimensions, linking harmonic measure, winding, and conformal invariance.

Contribution

It provides the first exact joint multifractal spectrum f(alpha, lambda) for potential scaling and winding near conformally invariant curves, generalizing previous harmonic measure results.

Findings

Derived the joint multifractal spectrum f(alpha, lambda).

Established a scaling law relating f(alpha, lambda) to f(bar alpha).

Applicable to O(N), Potts models, and SLE_kappa.

Abstract

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 $H \sim r^{\alpha}$ while the curve logarithmically spirals with a rotation angle phi=lambda ln r. It obeys the scaling law f(\alpha,\lambda)=(1+\lambda^2) f(\bar \alpha)-b\lambda^2 with \bar \alpha=\alpha/(1+\lambda^2) and b=(25-c)/{12}$, and where f(\alpha)\equiv 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 {\rm SLE}_{\kappa}.