Critical Exponents near a Random Fractal Boundary

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 $λ_n = 1/48 (\sqrt{1+24n\xt}+11)(\sqrt{1+24n\xt}-1)$. 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.

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

References in corpus (1)

• Two-Dimensional Copolymers and Exact Conformal Multifractality

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