Critical exponents near a random fractal boundary

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 characterized by the surface scaling dimension (x) over tilde. 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 characterized by a set of multifractal boundary exponents, given exactly by conformal invariance arguments to be lambda(n) = 1/48(root 1 + 24n (x) over tilde + 11)(root 1 + 24n (x) over tilde - 1). This result may be interpreted in terms of a scale-dependent distribution of opening angles a 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.