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Limit Profile for the high-temperature Curie-Weiss model

Lazaros Karageorgiou, Kyprianos-Iason Prodromidis

TL;DR

The paper analyzes Glauber dynamics for the high-temperature Curie–Weiss Ising model on the complete graph and proves a precise limit profile for the total-variation distance at the cutoff. It reduces the dynamics to a two-coordinate chain and shows an initial near-deterministic drift toward the center, followed by a diffusion regime described by a two-dimensional Ornstein–Uhlenbeck process. By performing a diffusion limit and explicit Gaussian-TV computations, the authors derive the limit profile Ψ(θ) as the TV distance between two Gaussians with the same covariance 1/(1-β) but shifted means, parameterized by θ and the limiting magnetization distribution. The approach combines drift analysis, coupling techniques, and diffusion-approximation theory, providing a robust framework that could extend to other mean-field or graph-structured models. This advances the understanding of limit profiles in mean-field dynamics and clarifies the structure of the cutoff in the Curie–Weiss setting.

Abstract

In this paper, we consider the Ising model on the complete graph, also known as the Curie-Weiss model, and establish the limit profile of the Glauber dynamics in the high-temperature regime. Our strategy is a two-dimensional analog of the method developed by Olesker-Taylor and Schmid for the Bernoulli-Laplace urn: The two-coordinate chain associated to the model evolves near-deterministically until just before the cutoff window, while afterwards it approximates a two-dimensional diffusion.

Limit Profile for the high-temperature Curie-Weiss model

TL;DR

The paper analyzes Glauber dynamics for the high-temperature Curie–Weiss Ising model on the complete graph and proves a precise limit profile for the total-variation distance at the cutoff. It reduces the dynamics to a two-coordinate chain and shows an initial near-deterministic drift toward the center, followed by a diffusion regime described by a two-dimensional Ornstein–Uhlenbeck process. By performing a diffusion limit and explicit Gaussian-TV computations, the authors derive the limit profile Ψ(θ) as the TV distance between two Gaussians with the same covariance 1/(1-β) but shifted means, parameterized by θ and the limiting magnetization distribution. The approach combines drift analysis, coupling techniques, and diffusion-approximation theory, providing a robust framework that could extend to other mean-field or graph-structured models. This advances the understanding of limit profiles in mean-field dynamics and clarifies the structure of the cutoff in the Curie–Weiss setting.

Abstract

In this paper, we consider the Ising model on the complete graph, also known as the Curie-Weiss model, and establish the limit profile of the Glauber dynamics in the high-temperature regime. Our strategy is a two-dimensional analog of the method developed by Olesker-Taylor and Schmid for the Bernoulli-Laplace urn: The two-coordinate chain associated to the model evolves near-deterministically until just before the cutoff window, while afterwards it approximates a two-dimensional diffusion.
Paper Structure (11 sections, 22 theorems, 148 equations)

This paper contains 11 sections, 22 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Definitions and main result
  3. Reduction to the two-coordinate chain
  4. Analysis of the two-coordinate chain
  5. Magnetization chain
  6. First part of the process
  7. Two useful lemmas
  8. Diffusion Approximation and TV distances
  9. Convergence to a Diffusion
  10. TV distance of Gaussians
  11. Proof of Theorem \ref{['main-theorem']}

Key Result

Theorem 1

For any $\theta\in\mathbb{R}$, let $t_{n,\theta}=(2(1-\beta))^{-1}\log(n)+\theta$. Then, where is a real constant depending only on $\beta\in[0,1)$.

Theorems & Definitions (43)

  • Definition 1.1: Mixing Time
  • Definition 1.2: Cutoff and Limit Profile
  • Definition 1.3
  • Theorem 1
  • Lemma 1.4
  • proof
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 33 more