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

Stein Andreas Bethuelsen, Malin Palö Forsström

TL;DR

The paper provides a general framework to test whether Poisson representable (PR) processes can capture classical lattice models like the Ising model and the contact process. By proving a mixing characterization for PR processes with finite-set intensity, the authors show that in dimensions $d\geq 2$ the Ising plus state on $\mathbb{Z}^2$ and the upper invariant measure of the supercritical CP are not PR, answering open questions about the reach of PR. They further prove that non-extremal translation-invariant Ising states cannot be Poisson representable in $\mathbb{Z}^d$ for $d\geq 2$, and establish an $L^2$-convergence result for ergodic averages of PR processes, which implies constraints on bimodality. Additionally, they demonstrate that while the CP upper invariant measure fails spatial mixing, it exhibits a directional mixing property, enriching the understanding of mixing phenomena in these models. Altogether, the work delineates the limitations of the PR class in representing key phase-transition regimes of Ising and CP systems and provides tools to analyze mixing structure via Finite-set intensity and restriction-type arguments.

Abstract

Forsström 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].

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

TL;DR

The paper provides a general framework to test whether Poisson representable (PR) processes can capture classical lattice models like the Ising model and the contact process. By proving a mixing characterization for PR processes with finite-set intensity, the authors show that in dimensions the Ising plus state on and the upper invariant measure of the supercritical CP are not PR, answering open questions about the reach of PR. They further prove that non-extremal translation-invariant Ising states cannot be Poisson representable in for , and establish an -convergence result for ergodic averages of PR processes, which implies constraints on bimodality. Additionally, they demonstrate that while the CP upper invariant measure fails spatial mixing, it exhibits a directional mixing property, enriching the understanding of mixing phenomena in these models. Altogether, the work delineates the limitations of the PR class in representing key phase-transition regimes of Ising and CP systems and provides tools to analyze mixing structure via Finite-set intensity and restriction-type arguments.

Abstract

Forsström et al. [8] recently introduced a large class of -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 nor the plus state of the Ising model on within the phase transition regime is Poisson representable. Moreover, we show that on , , any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in [8].
Paper Structure (10 sections, 15 theorems, 116 equations)

This paper contains 10 sections, 15 theorems, 116 equations.

Table of Contents

  1. Introduction, main results and outline of the paper
  2. Proofs of Theorem \ref{['theorem:Ising2']} and Theorem \ref{['theorem:cp1']}
  3. Mixing for Poisson representable processes
  4. (Lack of) mixing for the contact process and the Ising model
  5. Proof of Theorem \ref{['theorem: all middle Ising']}
  6. Impossibility of bimodality for Poisson representable processes
  7. Proof of Theorem \ref{['theorem: variance and limit']}
  8. Mixing properties for the contact process
  9. The upper invariant measure of the contact process is not spatially mixing
  10. The upper invariant measure is directional mixing

Key Result

Theorem 1.1

Let $X \sim \mu_{\beta}^+$ on $\{0,1\}^{\mathbb{Z}^2}$. For all $\beta > \beta_c$ it holds that $X \notin \mathcal{R}(\mathbb{Z}^2)$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lemma: transfer to limit new']}
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • proof : Proof of Lemma \ref{['lem:PRmain']}
  • Remark 2.1
  • ...and 28 more