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

2025/12/05 by Garoni, Timothy M., Perez, Aram, Zhou, Zongzheng
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.05340

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.

Citations

• Bounding Kolmogorov distances through Wasserstein and related integral probability metrics
• New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems
• Multivariate approximations in Wasserstein distance by Stein's method and Bismut's formula
• Fundamentals of Stein's method
• Stein's method for dependent random variables occurring in Statistical Mechanics
• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)
• A Derivation of n-Dimensional Spherical Coordinates

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• Stein's method for dependent random variables occurring in Statistical Mechanics
• Rescaled Magnetization for Critical Bipartite Mean-Fields Models
• Mean-field spin models -- Fluctuation of the magnetization and maximum likelihood estimator
• Non-uniform bounds for non-normal approximation via Stein's method with applications to the Curie--Weiss model and the imitative monomer-dimer model