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 $n\to \infty$ 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.

13 pages, 2 figures

References in corpus (3)

• Type III and spectral triples
• Quantum Statistical Mechanics, L-series and Anabelian Geometry
• Zeta functions that hear the shape of a Riemann surface

Cited by in corpus (6)

• $p$-Adic Mathematical Physics: The First 30 Years
• Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
• Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariants
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Quantum statistical mechanics in arithmetic topology
• Modular spectral triples and deformed Fredholm modules