Conformally Invariant Fractals and Potential Theory

Details Conformally Invariant Fractals and Potential Theory Conformally Invariant Fractals and Potential Theory

1999-08-22

arxiv.org/abs/cond-mat/9908314v1

Phys.Rev.Lett.84:1363-1367,2000

Physics

Condensed Matter

Statistical Mechanics

5 pages, 1 figure

Scientific paper

10.1103/PhysRevLett.84.1363

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 $D_{\rm EP} =sup_{\theta}\hat f(\theta)$ 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.

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