On stochastic generation of ultrametrics in high-dimension Euclidean spaces

Abstract

The proof of the theorem, which states that the Euclidean metric on the set of random points in an $n$-dimensional Euclidean space with the distribution of a special class, converges in probability in the limit $n\rightarrow\infty$ to the ultrametric is presented. The values of the ultrametric distance matrix is completely determined by variances of point coordinates. Probabilistic algorithm for the generation of finite ultrametric structures of any topology in high-dimensional Euclidean space is presented. The validity of the algorithm is demonstrated by explicit calculations of distance matrices with fixed dimensions and ultrametricity indexes for various dimensions of Euclidean space.