Emergence of patterns in random processes. III. Clustering in higher dimensions.

Newman et al. [Phys. Rev. E 86, 026103 (2012)10.1103/PhysRevE.86.026103] showed that points uniformly distributed as independent and identically distributed random variables with nearest-neighbor interactions form clusters with a mean number of three points in each. Here, we extend our analysis to higher dimensions, ultimately going to infinite dimensions, and we show that the mean number of points per cluster rises monotonically with a limiting value of four.