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

Newman et al. [Phys. Rev. E 86, 026103 (2012)] 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.

Figure 1

Random ensemble of points in two dimensions with associated directed graphs and cluster formation. The integers show the “indegree” of each vertex, i.e., the number of other points that identify with it as a nearest neighbor; the “outdegree” of each vertex is, due to our rules of clustering, automatically one.

Figure 2

Geometry associated with Bayesian probability calculation. First point at 1 showing radius $r$ of second point at 2 and the circles $A$ and $B$ plus radial lines relevant to analysis.

Figure 3

Venn diagram illustrating areas over which integration is performed. In the first case, it is all the area enclosed by $A$, namely the component in white and in light gray. In the second case, it is only the area in dark gray corresponding to $B$ less the area in the lens.

Figure 4

Demonstration that ${\widehat{V}}_n/V_n$ converges geometrically to zero.

Figure 5

Geometric configurations of clusters.