Conformally Invariant Fractals and Potential Theory

The multifractal (MF) distribution of the electrostatic potential near any conformally invariant fractal boundary, like a critical $O\left(N\right)$ loop or a $Q$-state Potts cluster, is solved in two dimensions. The dimension $\hat{f}\left(\theta \right)$ of the boundary set with local wedge angle $\theta$ is $\hat{f}\left(\theta \right)=\frac{\pi }{\theta }-\frac{25-c}{12}\frac{(\pi -\theta {)}^2}{\theta (2\pi -\theta )}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\mathrm{EP}}$ of the external perimeter and $D_{\mathrm{H}}$ of the hull of a Potts cluster obey the duality equation $(D_{\mathrm{EP}}-1\mathrm{})(D_{\mathrm{H}}-1\mathrm{})=\frac{1}{4}$. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.