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Low temperature asymptotic expansion for classical $O(N)$ vector models

Alessandro Giuliani, Sébastien Ott

TL;DR

The paper establishes that low-temperature expansions for multipoint spin correlations in classical $O(N)$ vector models with $d\ge 3$ are asymptotic series with explicitly controlled remainders. By reformulating the Gibbs measure in Gaussian coordinates and performing a systematic, mass-regularized Gaussian integration by parts, the authors derive an inductive algorithm yielding computable coefficients and strong remainder bounds. This yields a second-order magnetization formula and, in the $\beta\to\infty$ limit, a Gaussian Free Field description for the first $N-1$ spin components, highlighting a direct bridge between low-temperature spin correlations and free-field behavior. The approach provides a self-contained, non-multiscale proof that extends the BFLLS framework to non-abelian symmetry and non-gradient observables, complementing more intricate analyses like Balaban’s multiscale program.

Abstract

We consider classical $O(N)$ vector models in dimension three and higher and investigate the nature of the low-temperature expansions for their multipoint spin correlations. We prove that such expansions define asymptotic series, and derive explicit estimates on the error terms associated with their finite order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from reflection positivity and the infrared bound, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. Our method generalizes an approach, proposed originally by Bricmont and collaborators [6] in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables.

Low temperature asymptotic expansion for classical $O(N)$ vector models

TL;DR

The paper establishes that low-temperature expansions for multipoint spin correlations in classical vector models with are asymptotic series with explicitly controlled remainders. By reformulating the Gibbs measure in Gaussian coordinates and performing a systematic, mass-regularized Gaussian integration by parts, the authors derive an inductive algorithm yielding computable coefficients and strong remainder bounds. This yields a second-order magnetization formula and, in the limit, a Gaussian Free Field description for the first spin components, highlighting a direct bridge between low-temperature spin correlations and free-field behavior. The approach provides a self-contained, non-multiscale proof that extends the BFLLS framework to non-abelian symmetry and non-gradient observables, complementing more intricate analyses like Balaban’s multiscale program.

Abstract

We consider classical vector models in dimension three and higher and investigate the nature of the low-temperature expansions for their multipoint spin correlations. We prove that such expansions define asymptotic series, and derive explicit estimates on the error terms associated with their finite order truncations. The result applies, in particular, to the spontaneous magnetization of the 3D Heisenberg model. The proof combines a priori bounds on the moments of the local spin observables, following from reflection positivity and the infrared bound, with an integration-by-parts method applied systematically to a suitable integral representation of the correlation functions. Our method generalizes an approach, proposed originally by Bricmont and collaborators [6] in the context of the rotator model, to the case of non-abelian symmetry and non-gradient observables.
Paper Structure (26 sections, 23 theorems, 177 equations, 2 figures)

This paper contains 26 sections, 23 theorems, 177 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of the model and finite-volume infrared bound.
  3. Construction of the infinite volume measure and infrared bound.
  4. Main results
  5. Organization of the paper
  6. Notations
  7. Gaussian coordinates and the formal low-temperature expansion
  8. The Gibbs measure as a perturbed Gaussian
  9. The formal low temperature expansion
  10. The finite order truncations of the low temperature expansion
  11. Applications of the infrared bound
  12. Large deviation estimates on the spin correlations.
  13. Bounds on the moments of the $(u,\theta)$ coordinates
  14. Taylor expansion of the spin correlations
  15. Integration by part
  16. ...and 11 more sections

Key Result

Proposition 1.1

For any $h,\beta\geq 0$, any $L$, and any $f_1,\cdots,f_d :\Lambda_L\to\mathbb{R}^{N}$, In particular, for $f$ such that $\sum_{x}f(x) =0$, letting $f_k(x)=(-\Delta_L)^{-1}\nabla^{\hat{\mathrm{e}}_k}_x f$ in the previous equation, one has:

Figures (2)

  • Figure 1: $n=4$, $r_1=r_2=r_3=1$, $r_4=2$. Left: the "vertices" with the outgoing half-edges. Right: a possible pairing of the elements with $0,0'$ not merged.
  • Figure 2: $n=4$, $r_1=r_2=r_3=1$, $r_4=2$. Left: a possible pairing of the elements with $0,0'$ merged. Right: a path extraction.

Theorems & Definitions (47)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • ...and 37 more