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 characterised by the surface scaling dimension x̃. 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 λn = 1 48( √ 1 + 24nx̃ + 11)( √ 1 + 24nx̃ − 1). This result may be interpreted in terms of a scale-dependent distribution of opening angles α of the fractal boundary: on short distance scales these are sharply peaked around α = π/3. The subject of boundary critical behaviour [1] is by now well understood, particularly in two dimensions [2]. The two-point correlation function 〈φ(r)φ(R)〉 of a scaling operator φ, which behaves in the bulk at large distances at the critical point as |r−R|−2x, where x is the bulk scaling dimension of φ, is modified when one of the points (say r) is close to the boundary to the form 〈φ(r)φ(R)〉 ∼ |r||R||R/r|, (1) where x̃ is the corresponding boundary scaling dimension, and the angular dependence has been suppressed for clarity. In two dimensions, the role played by x̃ is emphasised by making the conformal mapping z → ln z of the upper half plane to a strip of width π: in that geometry the correlation function decays exponentially along the strip with an inverse correlation length equal to x̃ [3]. Eq. 1 refers to the case when the boundary is smooth (at least on scales ≪ r) and it is an interesting to ask whether these results are modified when the boundary is a fractal on these scales. The example of an edge (or corner in two dimensions) on the boundary was analysed some time ago [4] and it was shown that new edge scaling dimensions arise which depend continuously on the opening angle α. In two dimensions [5] this dependence is given by conformal invariance arguments by the simple form x(α) = πx̃/α. This suggests that close to a fractal boundary, which may be thought of as presenting a distribution of opening angles (which perhaps also depends on the scale at