Mixing for Poisson representable processes and consequences for the Ising model and the contact process

TL;DR

This paper investigates Poisson representable processes, establishing that key models like the contact process and Ising model in certain regimes are not Poisson representable, thus clarifying the class's limitations.

Contribution

It introduces new mixing-based criteria for Poisson representability and demonstrates these criteria exclude important models from the class.

Findings

The upper invariant measure of the supercritical contact process is not Poisson representable.

The plus state of the Ising model in the phase transition regime is not Poisson representable.

Non-extremal translation invariant states of the Ising model are not Poisson representable.

Abstract

Forsstr\"om et al. [8] recently introduced a large class of $\{0,1\}$-valued processes that they named Poisson representable. In addition to deriving several interesting properties for these processes, their main focus was determining which processes are contained in this class. In this paper, we derive new characteristics for Poisson representable processes in terms of certain mixing properties. Using these, we argue that neither the upper invariant measure of the supercritical contact process on $\mathbb{Z}^d$ nor the plus state of the Ising model on $\mathbb{Z}^2$ within the phase transition regime is Poisson representable. Moreover, we show that on $\mathbb{Z}^d$, $d\geq 2$, any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in [8].