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Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature

Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi

Abstract

The truncated two-point function of the ferromagnetic Ising model on $\mathbb Z^d$ ($d\ge3$) in its pure phases is proven to decay exponentially fast throughout the ordered regime ($β>β_c$ and $h=0$). Together with the previously known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: $(β,h) = (β_c,0)$.

Exponential decay of truncated correlations for the Ising model in any dimension for all but the critical temperature

Abstract

The truncated two-point function of the ferromagnetic Ising model on () in its pure phases is proven to decay exponentially fast throughout the ordered regime ( and ). Together with the previously known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point: .

Paper Structure

This paper contains 23 sections, 14 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Exponential decay of truncated correlations of the Ising model
  3. Mixing of the Fortuin-Kasteleyn representation of the Ising model
  4. Idea of the proof
  5. Open problems
  6. Organization
  7. Acknowledgments
  8. Background
  9. The FK-Ising model
  10. Spatial Markov property.
  11. Stochastic ordering for $q\ge1$.
  12. Edwards-Sokal coupling
  13. Griffiths inequality
  14. The random-current representation
  15. Applications of Pisztora's coarse-grain approach
  16. ...and 8 more sections

Key Result

Theorem \oldthetheorem

For the nearest-neighbor Ising model on $\mathbb{Z}^d$ in dimension $d \geq 3$, for any $\beta>\beta_c$ there exists $c = c(\beta,d) > 0$ such that for every $x,y\in\mathbb{Z}^d$,

Figures (3)

  • Figure 1: A realization of the event $\mathcal{H}_t$. The grey boxes are the bad blocks in $C_t$ that are adjacent to $C_{t + 1} \setminus C_t$. The dark rectangles represent the edges in $D_t$.
  • Figure 2: A schematic version of Lemma \ref{['lem:gluing']}. The thick lines represent the edges in the multigraph underlying $\mathbf n_1 + \mathbf n_2$ whereas the thin line represents the path $\Pi_{\bf B}=(v_0,\dots,v_k)$. We also depicted the sets $T_{\bf B}$ (filled squares) and $S_{\bf B} \setminus T_{\bf B} \cup \{v_k\}$ (crosses).
  • Figure 3: A simple representation of the set $V(\mathbf n_1, \mathbf n_2)$.

Theorems & Definitions (30)

  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem: Exponential mixing
  • proof : Proof of Theorem \ref{['thm:main']}
  • Corollary \oldthetheorem: Ratio weak mixing
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • proof : Proof of Lemma \ref{['lem:00']}
  • Lemma \oldthetheorem
  • proof
  • ...and 20 more