Finitary codings and stochastic domination for Poisson representable processes

TL;DR

This paper investigates the conditions under which certain random sets, constructed from independent finite subsets of integers, can be represented as finitary factors of IID processes and their stochastic domination properties, providing answers to open questions.

Contribution

It establishes that random sets formed from pairs are finitary factors of IID processes under exponential moment conditions and explores their stochastic domination and phase transition properties.

Findings

Random sets from pairs are finitary factors of IID processes with exponential moments.

Existence of exponential moments is necessary for such representations.

Random sets are stochastically dominated by Bernoulli percolation if and only if exponential moments exist.

Abstract

Construct a random set by independently selecting each finite subset of the integers with some probability depending on the set up to translations and taking the union of the selected sets. We show that when the only sets selected with positive probability are pairs, such a random set is a finitary factor of an IID process, answering a question of Forsstr\"om, Gantert and Steif. More generally, we show that this is the case whenever the distribution induced by the size of the selected sets has sufficient exponential moments, and that the existence of some exponential moment is necessary. We further show that such a random set is stochastically dominated by a non-trivial Bernoulli percolation if and only if there is a finite exponential moment, thereby partially answering another question of Forsstr\"om et al. We also give a partial answer to a third question regarding a form of phase transition. These results also hold on $\mathbb{Z}^d$ with $d \ge 2$. In the one-dimensional case, under the condition that the distribution induced by the diameter of the selected sets has an exponential moment, we further show that such a random set is finitarily isomorphic to an IID process.