Conformally Invariant Fractals and Potential Theory
Résumé
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 $\hat f(\theta)$ of the boundary set with local wedge angle $\theta$ is $\hat f(\theta)={\pi}/{\theta} - {25-c}/{12} {(\pi-\theta)^2}/{\theta(2\pi-\theta)}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\rm EP}$ %=sup_{\theta}\hat f(\theta)$ of the external perimeter and $D_{\rm H}$ of the hull of a Potts cluster obey the duality equation $(D_{\rm EP}-1)(D_{\rm H}-1)={1}/{4}$. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.