## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

Motivated by Murtagh's experimental observation that sparse random samples of the hypercube become more and more ultrametric as the dimension increases, we prove that his ultrametricity index as well as our topological ultrametricity index converges in probability to one as dimension increases, if the sample size remains fixed. This is done for uniformly and normally distributed samples in the Euclidean hypercube, and for uniformly distributed samples in $\mathds{F}_2^N$ with Hamming distance, as well as for uniform random categorial data in complete disjunctive form. A second result is that both ultrametricity indices vanish in the limit for the full hypercube $\mathds{F}_2^N$ as dimension $N$ increases, whereby Murtagh's ultrametricity index is greater than the topological ultrametricity index if $N$ is large.