  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.
