Random walks, self-avoiding walks and 1/f noise on finite lattices

We investigate the probabilities for a return to the origin at step n of a random walker on a finite lattice. As a consistent measure only the first returns to the origin appear to be of relevance; these include paths with self-intersections and self-avoiding polygons. Their return probabilities are power-law distributed giving rise to Ilf (b) noise. Most striking is the behavior of the self-avoiding polygons exhibiting a slope b = 0.83 for d = 2 and b = 0.93 for d = 3 independent on lattice structure.