Conformally Invariant Fractals and Potential Theory

Abstract

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 $\hat f(θ)$ of the boundary set with local wedge angle $θ$ is $\hat f(θ)=\fracπθ -\frac{25-c}{12} \frac{(π-θ)^2}{θ(2π-θ)}$, with $c$ the central charge of the model. As a corollary, the dimensions $D_{\rm EP} =sup_θ\hat f(θ)$ 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.