  Motivated by Alain-Sol Sznitman's interlacement process, we consider the set
of $\{0,1\}$-valued processes which can be constructed in an analogous way,
namely as a union of sets coming from a Poisson process on a collection of
sets. Our main focus is to determine which processes are representable in this
way. Some of our results are as follows. (1) All positively associated Markov
chains and a large class of renewal processes are so representable. (2) Whether
an average of two product measures, with close densities, on $n$ variables, is
representable is related to the zeroes of the polylogarithm functions. (3)
Using (2), we show that a number of tree indexed Markov chains as well as the
Ising model on $ \mathbb{Z}^d ,$ $ d\geq 2,$ for certain parameters are not so
representable. (4) The collection of permutation invariant processes which are
representable corresponds exactly to the set of infinitely divisible random
variables on $[0,\infty]$ via a certain transformation. (5) The supercritical
(low temperature) Curie-Weiss model is not representable for large $n$.
