Generalized Divide and Color models

In this paper, we initiate the study of "Generalized Divide and Color Models". A very special interesting case of this is the "Divide and Color Model" (which motivates the name we use) introduced and studied by Olle H\"aggstr\"om. In this generalized model, one starts with a finite or countable set $V$, a random partition of $V$ and a parameter $p\in [0,1]$. The corresponding Generalized Divide and Color Model is the $\{0,1\}$-valued process indexed by $V$ obtained by independently, for each partition element in the random partition chosen, with probability $p$, assigning all the elements of the partition element the value 1, and with probability $1-p$, assigning all the elements of the partition element the value 0. Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have? The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model, (3) the stationary distributions for the Voter Model, (4) random walk in random scenery and of course (5) the original Divide and Color Model.