Tracing a diffusion-limited aggregate: Self-affine versus self-similar scaling

The geometry of diffusion-limited aggregation clusters is mapped into single-valued functions by tracing the surface of the aggregate and recording the X (or Y) coordinate of the position of a walker moving along the perimeter of the cluster as a function of the arc length. Our numerical results and scaling arguments show that the related plots can be considered as self-affine functions whose scaling behavior is determined by the exponent H=1/D, where D is the fractal dimension of the aggregates.