Coloring percolation clusters at random.
A particular way of random coloring of graph vertices is considered. For this purpose, \(r\) different colors are allowed, denoted by \(0,1,\dots ,r-1\). In the first step of the procedure, edges of the graph are deleted independently at random with a fixed probability \(1-p\). The resulting object is in percolation theory known as i.i.d.\ (or Bernoulli) bond percolation. In the second step, for each component of the obtained subgraph the same color is assigned to all its vertices. This color is chosen at random according to some fixed probability distribution and independently for different connected components. The whole procedure is called ``divide and color model'' and the main aim of the paper is to study its properties in the particular case of cubic lattice graph \(Z^d\). First, the percolation properties of the model are investigated. That means the conditions for existence of an infinite connected component are given. The percolation substantially depends on the value of the parameter \(p\), and, to some extend, also to the dimension \(d\). Sharpest results are obtained for \(d=2\) with a relation to similar results for the two-dimensional Ising model. Further, as the second main result of the paper, it is proved that the model under consideration does not obey the Markov property of any order. But, again only for some values of \(p\) it is shown that the model is quasilocal which is equivalent to the Gibbsian property.
scientific article; zbMATH DE number 2138911