The rate of convergence of the critical mean-field O(N) magnetization via multivariate nonnormal Stein's method

TL;DR

The paper analyzes the critical mean-field $O(N)$ spin model ($N\ge 2$) and provides a quantitative rate of convergence for the finite-volume magnetization to its nonnormal limiting law at criticality. By extending a multivariate nonnormal Stein's method framework and constructing the Stein solution via the overdamped Langevin semigroup, the authors derive explicit bounds showing the Wasserstein distance between $n^{-3/4}S_n$ and the limit with density proportional to $\exp(-a_N|x|^4)$ scales as $\mathcal{O}(n^{-1/2})$ for all $N\ge 2$. They develop rigorous SDE/semigroup machinery, including gradient bounds via Beltrami–Elworthy–Li formulas and precise control of variation processes, to handle the multivariate nonnormal limit. The results generalize known univariate and Curie–Weiss-type convergence to the multivariate $O(N)$ setting and provide a robust framework for quantitative distributional approximations at criticality in mean-field spin models.

Abstract

We study the distribution of the magnetization of the critical mean-field O(N) model with N > 1. Specifically, we bound the Wasserstein distance between the finite-volume and limiting distributions, in terms of the number of spins. To achieve this, we extend a recent multivariate nonnormal approximation theorem. This generalizes known results for the Curie-Weiss magnetization to the multivariate O(N) setting.

Theorems & Definitions (31)

• Theorem 1.1: KirkpatrickMeckes2013KirkpatrickNawaz2016FangKoike2022
• Theorem 1.2
• Remark 1.3
• Theorem 2.2
• Theorem 2.4: FangShaoXu2019
• proof
• Proposition 3.1: DunlopNewman1975
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['Theorem: Critical O(N)']}.