Critical Exponents near a Random Fractal Boundary

TL;DR

This paper derives exact multifractal boundary exponents for correlation functions near a random fractal boundary in two dimensions, revealing scale-dependent boundary behavior influenced by conformal invariance.

Contribution

It introduces a set of exact multifractal boundary exponents for random fractal boundaries, extending conformal invariance results to complex boundary geometries.

Findings

Boundary correlation functions exhibit multifractal scaling.

Exact formulas for boundary exponents are derived.

Boundary angle distribution is scale-dependent, peaked at π/3.

Abstract

The critical behaviour of correlation functions near a boundary is modified from that in the bulk. When the boundary is smooth this is known to be characterised by the surface scaling dimension $\xt$. We consider the case when the boundary is a random fractal, specifically a self-avoiding walk or the frontier of a Brownian walk, in two dimensions, and show that the boundary scaling behaviour of the correlation function is characterised by a set of multifractal boundary exponents, given exactly by conformal invariance arguments to be $\lambda_n = 1/48 (\sqrt{1+24n\xt}+11)(\sqrt{1+24n\xt}-1)$. This result may be interpreted in terms of a scale-dependent distribution of opening angles $\alpha$ of the fractal boundary: on short distance scales these are sharply peaked around $\alpha=\pi/3$. Similar arguments give the multifractal exponents for the case of coupling to a quenched random bulk geometry.