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Emergence of fractional Gaussian free field correlations in subcritical long-range Ising models

Trishen S. Gunaratnam, Romain Panis

Abstract

We study corrections to the scaling limit of subcritical long-range Ising models with (super)-summable interactions on $\mathbb{Z}^d$. For a wide class of models, the scaling limit is known to be white noise, as shown by Newman (1980). In the specific case of couplings $J_{x,y}=|x-y|^{-d-\boldsymbolα}$, where $\boldsymbolα>0$ and $|\cdot|$ is the Euclidean norm, we find an emergence of fractional Gaussian free field correlations in appropriately renormalised and rescaled observables. The proof exploits the exact asymptotics of the two-point function, first established by Newman and Spohn (1998), together with the rotational symmetry of the interaction.

Emergence of fractional Gaussian free field correlations in subcritical long-range Ising models

Abstract

We study corrections to the scaling limit of subcritical long-range Ising models with (super)-summable interactions on . For a wide class of models, the scaling limit is known to be white noise, as shown by Newman (1980). In the specific case of couplings , where and is the Euclidean norm, we find an emergence of fractional Gaussian free field correlations in appropriately renormalised and rescaled observables. The proof exploits the exact asymptotics of the two-point function, first established by Newman and Spohn (1998), together with the rotational symmetry of the interaction.
Paper Structure (10 sections, 10 theorems, 81 equations)

This paper contains 10 sections, 10 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Convergence to white noise
  3. Triviality of any limit point
  4. Computation of the covariance kernel
  5. Proof of white noise convergence
  6. Emergence of fractional GFF correlations
  7. Case $\boldsymbol{\alpha}\in \mathbb{R}_{> 0} \setminus 2\mathbb{N}^*$
  8. Case $\boldsymbol{\alpha}\in 2\mathbb{N}^*$
  9. Proof of emergence of fractional GFF correlations
  10. Rotational symmetry and Taylor's theorem

Key Result

Proposition 1.1

Let $d\geq 2$, $\boldsymbol{\alpha},\mathbf{C}> 0$ in the interaction eq: interaction, and $\beta<\beta_c$. Then, there exists $\delta > 0$ such that for every $x\in \mathbb Z^d\setminus \lbrace 0\rbrace$, where $\chi(\beta):=\sum_{x\in \mathbb Z^d}\langle \sigma_0\sigma_x\rangle_\beta$. As an immediate consequence, we obtain polynomial decay of correlations: there exists $C=C(\beta,J)>0$ such th

Theorems & Definitions (29)

  • Proposition 1.1: newman1998shibaAo
  • Definition 1.2: Smeared observable
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6: Convergence to the white noise
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 19 more