Angular structure of lacunarity, and the renormalization group

We formulate the angular structure of lacunarity in fractals, in terms of a symmetry reduction of the three point correlation function. This provides a rich probe of universality, and first measurements yield new evidence in support of the equivalence between self-avoiding walks (SAW's) and percolat ion perimeters in two dimensions. We argue that the lacunarity reveals much of the renormalization group in real space. This is supported by exact cal culations for random walks and measured data for percolation clusters and S AW's. Relationships follow between exponents governing inward and outward p ropagating perturbations. and we also find a very general test for the cont ribution of long-range interactions.