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

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.