Conformally Invariant Fractals and Potential Theory

TL;DR

This paper derives the multifractal distribution of electrostatic potential near conformally invariant fractal boundaries in two dimensions, revealing new formulas for boundary dimensions and duality relations in critical models.

Contribution

It provides an exact solution for the multifractal potential distribution near conformally invariant fractals and establishes duality relations for boundary dimensions in critical statistical models.

Findings

Derived the multifractal spectrum of electrostatic potential near fractal boundaries.

Established a duality relation between external perimeter and hull dimensions.

Obtained a covariant multifractal spectrum for self-avoiding walks at cluster boundaries.

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