Spacing distributions for point processes on a regular fractal

Abstract

The homogeneous Poisson point process in Rd (denoted by Pd) is a basic model of stochastic geometry and modern statistical physics. Using ideas from fractal geometry, geometrical statistics, and random matrix theory, we introduce the model of random points on a self-similar fractal as a model of intermediate statistics, in the sense that the interpoint spacing statistics of the model are intermediate between those of P1 and P2 when the fractal dimension is in between 1 and 2, and intermediate between those of P2 and P3 when the fractal dimension is in between 2 and 3, and so on. We also introduce the idea of using a continuous family of such models to interpolate between P1 and P2 and thereby effectuate crossover transitions between P1 statistics and P2 statistics. We first derive the kth-nearest-neighbor spacing distribution for the general model, and then study the interpoint spacing statistics of several realizations of the model involving Sierpinski fractals in R2 and R3. We also study a realization of a continuous interpolation between P1 and P2, in particular a continuous interpolation between a point process on a line and a point process on a plane-filling curve, using the continuous family of self-similar Koch curves in R2. In the latter study, we specifically analyze the second-nearest-neighbor interpoint spacing statistics, which undergo a crossover transition between semi-Poisson and Ginibre statistics.