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Central limit theorem for high temperature Ising models via martingale embedding

Xiao Fang, Yi-Kun Zhao

TL;DR

The paper develops a martingale-embedding framework to prove a central limit theorem for projections $W_n=\theta^T X$ of high-dimensional Ising models under a high-temperature Poincaré inequality, providing a non-asymptotic $\mathcal{W}_2$ bound in terms of two-point and three-point covariances. The method represents $W_n$ via a Föllmer-process-based martingale and compares it to a Gaussian, with the bound depending on covariances computed under varying external fields. It applies the result to two Ising-model regimes: (i) finite-range interactions on a lattice yielding CLT for the total magnetization, and (ii) ferromagnetic models under Dobrushin’s condition with asymptotic normality after standardization when the external field is bounded. The approach extends to continuous vectors obeying a Poincaré inequality and offers a novel CLT route beyond spatially structured models. Altogether, it provides a covariance-type bound and a new high-dimensional CLT paradigm for dependent vectors under a Poincaré framework.

Abstract

We use martingale embeddings to prove a central limit theorem (CLT) for projections of high-dimensional random vectors satisfying a Poincaré inequality. We obtain a non-asymptotic error bound for the CLT in 2-Wasserstein distance involving two-point and three-point covariances. We present two illustrative applications to Ising models: one with finite-range interactions and the other in the ferromagnetic case under the Dobrushin condition.

Central limit theorem for high temperature Ising models via martingale embedding

TL;DR

The paper develops a martingale-embedding framework to prove a central limit theorem for projections of high-dimensional Ising models under a high-temperature Poincaré inequality, providing a non-asymptotic bound in terms of two-point and three-point covariances. The method represents via a Föllmer-process-based martingale and compares it to a Gaussian, with the bound depending on covariances computed under varying external fields. It applies the result to two Ising-model regimes: (i) finite-range interactions on a lattice yielding CLT for the total magnetization, and (ii) ferromagnetic models under Dobrushin’s condition with asymptotic normality after standardization when the external field is bounded. The approach extends to continuous vectors obeying a Poincaré inequality and offers a novel CLT route beyond spatially structured models. Altogether, it provides a covariance-type bound and a new high-dimensional CLT paradigm for dependent vectors under a Poincaré framework.

Abstract

We use martingale embeddings to prove a central limit theorem (CLT) for projections of high-dimensional random vectors satisfying a Poincaré inequality. We obtain a non-asymptotic error bound for the CLT in 2-Wasserstein distance involving two-point and three-point covariances. We present two illustrative applications to Ising models: one with finite-range interactions and the other in the ferromagnetic case under the Dobrushin condition.

Paper Structure

This paper contains 4 sections, 8 theorems, 87 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of \ref{['thm:1']}
  3. Proof of \ref{['thm:2']}
  4. Proof of \ref{['thm:3']}

Key Result

Theorem 1.1

Let $X$ follow the Ising model eq:Isingmodel with $0\preceq A\prec I_n$. Let $\theta=(\theta_1,\dots, \theta_n)^\top\in \mathbb{R}^n$ be a unit vector. Let Then, we have, for any $\varepsilon<1/2$, where $N(\mu_n, \sigma_n^2)$ denotes the normal distribution with mean $\mu_n$ and variance $\sigma_n^2$, the supremum in eq:thm1 is over all possible external fields $h\in \mathbb{R}^n$, and the exp

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.1: Poincaré suffices for the result
  • Corollary 1.1
  • Corollary 1.2
  • Remark 1.2: Beyond quadratic interactions
  • Remark 1.3: CLT for projections of continuous random vectors
  • Remark 2.1
  • Lemma 2.1
  • proof : Proof of \ref{['lem:larget']}
  • Lemma 2.2
  • ...and 7 more