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

Area of Science:

• Statistical Physics
• Computational Physics
• Multidimensional Analysis

Background:

• Previous work demonstrated that uniformly distributed points with nearest-neighbor interactions form clusters of approximately three points in 2D.
• Understanding cluster formation in random point distributions is crucial for various scientific fields.

Purpose of the Study:

• To extend the analysis of cluster formation to higher and infinite dimensions.
• To determine how dimensionality affects the mean number of points per cluster.

Main Methods:

• Utilized mathematical analysis to extend the model to arbitrary dimensions.
• Investigated the behavior of point distributions in the limit of infinite dimensions.

Main Results:

• The mean number of points per cluster was shown to increase monotonically with dimensionality.
• A limiting value of four points per cluster was identified as the dimension approaches infinity.

Conclusions:

• The dimensionality of the space significantly influences the clustering of uniformly distributed points.
• The findings provide insights into the statistical properties of random point distributions in high-dimensional spaces.