Spontaneous clustering in Markov chains. I. Fractal Dust

The review is devoted to the description of the statistical properties of a set of isolated points randomly distributed in space, which are nodes of one (or a family of independent) realizations of a Markov chain. The purpose of the analysis of this model is to study the conditions for the emergence of clusters in the set of these nodes and to describe their characteristics. This (first) part of the review introduces the basic concepts of statistics of point distributions: generating functionals, many-particle densities, factorial moments, Markov chains, correlation functions. The part ends with a description of one-dimensional self-similar (in the statistical sense) sets generated by a Poisson-fractional random process and a demonstration of the clustering phenomenon.