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 cha racterized by the surface scaling dimension (x) over tilde. We consider the case when the boundary is a random fractal, specifically a self-avoiding w alk or the frontier of a Brownian walk, in two dimensions, and show that th e boundary scaling behaviour of the correlation function is characterized b y a set of multifractal boundary exponents, given exactly by conformal inva riance arguments to be lambda(n) = 1/48(root 1 + 24n (x) over tilde + 11)(r oot 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. Si milar arguments give the multifractal exponents for the case of coupling to a quenched random bulk geometry.