2004/02/16 by A. M. Vershik · 1 citation
Mathematics · #math.RT #math.GT #math.MG #math.PR #msc:28-XX #msc:60-XX #msc:51-XX
paper · pdf · doi:10.1070/rm2004v059n02abeh000718
38 PAGES
arxiv created 2004/02/16 · arxiv updated 2015/06/26
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particulary universal) space to the properties of distance matrices. We show the link between those questions and classification of the Polish spaces with measure (Gromov or metric triples) and with the problem about S∞-invariant measures in the space of symmetric matrices. One of the new effects -exsitence in Urysohn space so called anarchical uniformly distributed sequences. We give examples of other categories in which the randomness and universality coincide (graph, etc.).