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

TL;DR

This paper analyzes the convergence rate of the magnetization distribution in the critical mean-field O(N) model, providing bounds on the Wasserstein distance and extending multivariate nonnormal approximation techniques.

Contribution

It extends multivariate nonnormal Stein's method to the O(N) model, generalizing results from the Curie-Weiss case to higher dimensions.

Findings

Bounded Wasserstein distance between finite and limiting distributions

Extended nonnormal approximation theorem to multivariate O(N) models

Generalized convergence results from Curie-Weiss to O(N) setting

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.