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 $\hat{f}(\theta )$ of the boundary set with local wedge angle $\theta$ 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 $D_{\mathrm{EP}}=su{p}_{\theta }\hat{f}(\theta )$ of the external perimeter and $D_{\mathrm{H}}$ of the hull of a Potts cluster obey the duality equation $(D_{\mathrm{EP}}-1)(D_{\mathrm{H}}-1)=1/4$. A related covariant MF spectrum is obtained for self-avoiding walks anchored at cluster boundaries.

5 pages, 1 figure

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