Central limit theorem for high temperature spin models via martingale embedding

TL;DR

This paper proves a central limit theorem for high-dimensional spin models using martingale embeddings, providing explicit error bounds and applying to various models at high temperature.

Contribution

Introduces a novel martingale embedding approach to establish a CLT for high-dimensional spin models with explicit error bounds and broad applicability.

Findings

CLT holds for projections of high-dimensional spin vectors satisfying Poincaré inequality.

Provides non-asymptotic error bounds in 2-Wasserstein distance.

Demonstrates applicability to Ising, Dobrushin ferromagnetic, and Sherrington-Kirkpatrick models.

Abstract

We use martingale embeddings to prove a central limit theorem (CLT) for one-dimensional projections of high-dimensional random vectors in $\{-1,1\}^n$ satisfying a Poincar\'e inequality. We obtain a non-asymptotic error bound involving two-point and three-point functions for the CLT in 2-Wasserstein distance. We present three illustrative applications: Ising model with finite-range interactions, ferromagnetic Ising model under the Dobrushin condition, and the Sherrington-Kirkpatrick spin glass model at sufficiently high temperature. In all the examples, we allow heterogeneous external fields.