Poisson Representable Processes

Abstract

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 that 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~$n$.