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A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures

Rodrigo Bissacot, Henrique Corsini

TL;DR

The paper delivers a convergent low-temperature cluster expansion for the one-dimensional long-range Ising model with polynomial decay $J(r)=r^{-\alpha}$ for $\alpha\in(1,2]$, using Fröhlich-Spencer contours reformulated as a hard-core polymer gas. By constructing robust one-, two-, and many-vertex tree bounds and leveraging contour-to-site estimates, it proves absolute convergence of the pressure series for sufficiently large $\beta$ and establishes analyticity in a finite-field disk. A central result is that the two-point function decays algebraically with rate $\alpha$, and the work provides a framework to bound $n$-point correlations via polymer and contour expansions. Collectively, this advances the rigorous understanding of long-range 1D Ising models by removing perturbative constraints on nearest-neighbor interactions and showing that the decay exponent matches the interaction decay, with implications for phase structure and correlation phenomena in low dimensions.

Abstract

In this work, a convergent low-temperature cluster expansion of the one-dimensional long-range ferromagnetic Ising model with polynomial decay $α\in (1,2]$ is developed; that is, $J(r)=r^{-α}$. As an application, the $n$-point correlations are studied and the two-point correlation is shown to be algebraic with rate of decay exactly $α$.

A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures

TL;DR

The paper delivers a convergent low-temperature cluster expansion for the one-dimensional long-range Ising model with polynomial decay for , using Fröhlich-Spencer contours reformulated as a hard-core polymer gas. By constructing robust one-, two-, and many-vertex tree bounds and leveraging contour-to-site estimates, it proves absolute convergence of the pressure series for sufficiently large and establishes analyticity in a finite-field disk. A central result is that the two-point function decays algebraically with rate , and the work provides a framework to bound -point correlations via polymer and contour expansions. Collectively, this advances the rigorous understanding of long-range 1D Ising models by removing perturbative constraints on nearest-neighbor interactions and showing that the decay exponent matches the interaction decay, with implications for phase structure and correlation phenomena in low dimensions.

Abstract

In this work, a convergent low-temperature cluster expansion of the one-dimensional long-range ferromagnetic Ising model with polynomial decay is developed; that is, . As an application, the -point correlations are studied and the two-point correlation is shown to be algebraic with rate of decay exactly .
Paper Structure (27 sections, 35 theorems, 378 equations, 15 figures)

This paper contains 27 sections, 35 theorems, 378 equations, 15 figures.

Table of Contents

  1. Introduction
  2. A brief history of the one-dimensional long-range Ising model
  3. Description of the results
  4. Preliminaries
  5. The long-range one-dimensional ferromagnetic Ising model
  6. Configurations as collections of spin flips
  7. Long-range contours
  8. Crucial estimates
  9. Polymers
  10. A polymer gas associated to the long-range Ising model
  11. The formal representation of the pressure
  12. Upper bounds to the activity
  13. Sums over trees
  14. One-vertex estimates
  15. Two-vertex estimates
  16. ...and 12 more sections

Key Result

Theorem 1.1

For all $\alpha\in(1,2]$ and $\Lambda\Subset\mathbb{Z}$, there exist $\beta_0=\beta_0(\alpha)>0$ such that if $\beta\geq\beta_0$, the series converges absolutely. Furthermore, for all $A\subset\Lambda$ and $\beta\geq\beta_0$, there exists $r_0\coloneqq r_0(\beta,\lvert A\rvert)>0$ independent of $\Lambda$ such that the series is analytic in the polydisk $\mathbb{D}_{r_0}^A$, where $\mathbb{D}_{r

Figures (15)

  • Figure 1: All three contours are compatible, but $\gamma_1$ is not positively compatible with $\gamma_2$. On the other hand, $\gamma_3$ is positively compatible with both $\gamma_1$ and $\gamma_2$. Hence, $\Gamma=\{\gamma_1,\gamma_2,\gamma_3\}$ is not a positive collection.
  • Figure 2: Take $\Gamma$ as in Figure \ref{['fig: positive-compatibility']}, then $\partial\Gamma=\{\gamma_1,\gamma_3\}$.
  • Figure 3: An example of the procedure whereof one obtains the coarsest decomposition of $\Gamma\in\mathscr{E}$ into compatible polymers.
  • Figure 4: Another example of the procedure whereof one obtains the coarsest decomposition of $\Gamma\in\mathscr{E}$ into compatible polymers.
  • Figure 5: A concrete example of Notation \ref{['not: trees-with-set-of-leaves-fixed']} where $B=\{1,2,4\}$.
  • ...and 10 more figures

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.4: $M$-contour
  • Definition 2.5: $M$-partition
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • Definition 2.8
  • ...and 89 more