The modern theory of random partitions comes from theoretical population genetics (Ewens, 1972). From Ewens´ sampling formula emerged Kingman´s theory of exchangeable random partitions (Kingman, 1978) and, eventually, the coalescent process (Kingman, 1982). In the probability literature, further generalizations of Kingman´s theory has produced connections between partition-valued Markov processes, Brownian motion and stable subordinators (Bertoin, 2006; Pitman, 2005).

We discuss a general class of partition- valued Markov processes whose behavior we can describe precisely. Specifically, we give a Levy-Ito representation for exchangeable Feller processes on the space [k]^N of infinite k-colorings. In discrete-time, we can represent the evolution of these processes by a product of i.i.d. stochastic matrices. Using this representation, we can show the cut-off phenomenon for some Markov chains in this class.