On the ultrametric generated by random distribution of points in Euclidean spaces of large dimensions with correlated coordinates

TL;DR

This paper extends previous results showing that the normalized Euclidean distances among high-dimensional random points form an ultrametric structure, now including cases with weakly correlated coordinates, with illustrative examples provided.

Contribution

It generalizes the convergence of distance matrices to ultrametrics from independent to weakly correlated coordinates in high-dimensional spaces.

Findings

Distance matrices converge to ultrametrics as dimension grows

Ultrametrics are determined by conditional variances of coordinates

Includes examples of ultrametric space generation algorithms

Abstract

In a recent paper the author proved a theorem to the effect that the matrix of normalized Euclidean distances on the set of specially distributed random points in the $n$-dimensional Euclidean space $\mathbb R^{n}$ with independent coordinates converges in probability as $n\rightarrow\infty$ to an ultrametric matrix, the latter being completely determined by the expectations of conditional variances of random coordinates of points. The main theorem of the present paper extends this result to the case of weakly correlated coordinates of random points. Prior to formulating and stating this result we give two illustrative examples describing particular algorithms of generation of such ultrametric spaces.