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Approximate FKG inequalities for phase-bound spin systems

Satyaki Mukherjee, Vilas Winstein

Abstract

The FKG inequality is an invaluable tool in monotone spin systems satisfying the FKG lattice condition, which provides positive correlations for all coordinate-wise increasing functions of spins. However, the FKG lattice condition is somewhat brittle and is not preserved when confining a spin system to a particular phase. For instance, consider the Curie-Weiss model, which is a model of a ferromagnet with two phases at low temperature corresponding to positive and negative overall magnetization. It is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the model arise primarily from the global choice of positive or negative magnetization. In this article, we show that the individual phases do indeed satisfy an approximate form of the FKG inequality in a class of generalized higher-order Curie-Weiss models (including the standard Curie-Weiss model as a special case), as well as in ferromagnetic exponential random graph models (ERGMs). To cover both of these settings, we present a general result which allows for the derivation of such approximate FKG inequalities in a straightforward manner from inputs related to metastable mixing; we expect that this general result will be widely applicable. In addition, we derive some consequences of the approximate FKG inequality, including a version of a useful covariance inequality originally due to Newman as well as Bulinski and Shabanovich. We use this to extend the proof of the central limit theorem for ERGMs within a phase at low temperatures, due to the second author, to the non-forest phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini for the edge-triangle model.

Approximate FKG inequalities for phase-bound spin systems

Abstract

The FKG inequality is an invaluable tool in monotone spin systems satisfying the FKG lattice condition, which provides positive correlations for all coordinate-wise increasing functions of spins. However, the FKG lattice condition is somewhat brittle and is not preserved when confining a spin system to a particular phase. For instance, consider the Curie-Weiss model, which is a model of a ferromagnet with two phases at low temperature corresponding to positive and negative overall magnetization. It is not a priori clear if each phase internally has positive correlations for increasing functions, or if the positive correlations in the model arise primarily from the global choice of positive or negative magnetization. In this article, we show that the individual phases do indeed satisfy an approximate form of the FKG inequality in a class of generalized higher-order Curie-Weiss models (including the standard Curie-Weiss model as a special case), as well as in ferromagnetic exponential random graph models (ERGMs). To cover both of these settings, we present a general result which allows for the derivation of such approximate FKG inequalities in a straightforward manner from inputs related to metastable mixing; we expect that this general result will be widely applicable. In addition, we derive some consequences of the approximate FKG inequality, including a version of a useful covariance inequality originally due to Newman as well as Bulinski and Shabanovich. We use this to extend the proof of the central limit theorem for ERGMs within a phase at low temperatures, due to the second author, to the non-forest phase-coexistence regime, answering a question posed by Bianchi, Collet, and Magnanini for the edge-triangle model.
Paper Structure (22 sections, 21 theorems, 137 equations, 1 figure)

This paper contains 22 sections, 21 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Generalized Curie--Weiss models
  4. Exponential random graph models
  5. Results
  6. Proof ideas
  7. Outline of the article
  8. Acknowledgements
  9. A general theorem for deriving approximate FKG inequalities
  10. Tilted measure and dynamics
  11. Proofs of key bounds
  12. Application to generalized Curie--Weiss models
  13. Large deviations principle
  14. Contraction in almost the entire metastable well
  15. Proof of Theorem \ref{['thm:main_gcwm']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}_{\beta,m_*}^{\mathrm{GCW}}$ denote a phase measure in an $N$-spin generalized Curie--Weiss model, for some $m_* \in U^\mathrm{GCW}_\beta$, recalling these definitions from Section sec:intro_setup_gcwm, and consider a random spin configuration $X \sim \mathcal{M}_{\beta,m_*}^{\mathrm

Figures (1)

  • Figure 1: Left: even if $x$ and $y$ are in the metastable well, $x \wedge y$ or $x \vee y$ may not be. Right: to remedy this, we assume that $x$ and $y$ are almost ordered, which means that $x \wedge y$ is close to either $x$ or $y$, and similarly for $x \vee y$. To make this argument work, we also need to restrict $x$ and $y$ to be a bit away from the boundary of the metastable well, so that nearby points remain in the well. Note that in these pictures, we visualize the relation $x \leq y$ as having $x$ lie within the cone under $y$ with sides at 45-degree angles from the horizontal.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm:general']}
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lem:starting_point']}
  • Lemma 2.5
  • ...and 22 more