Conformally Invariant Fractals and Potential Theory

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