Topological and geometrical properties of random fractals

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Abstract
The underlying structure of random fractals is described in terms of two exponents, zeta and nu . The exponent zeta is sensitive only to the topology or 'connectedness' of the fractal, whereas nu depends only on the geometry of the fractal. The topological exponent arises from the scaling form for the distribution of paths, nl, on a finite fractal, where a path is defined as the shortest walk from one site to another. This path length distribution function is then used to derive expressions for the radius of gyration, pair correlation function, static structure factor and intensity of radiation scattered from percolation clusters at the gel point. From these calculations the fractal dimension is shown to be zeta / nu . Finally, recent numerical results for percolation clusters, percolation backbones, and lattice animals are discussed.