Critical Exponents near a Random Fractal Boundary

Details Critical Exponents near a Random Fractal Boundary Critical Exponents near a Random Fractal Boundary

1998-12-29

arxiv.org/abs/cond-mat/9812416v2

J.Phys. A32 (1999) L177

Physics

Condensed Matter

Statistical Mechanics

13 pages. Comments on relation to results in quenched random bulk added, and on relation to other recent work. Typos corrected

Scientific paper

10.1088/0305-4470/32/16/001

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.

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