Large N limit of the Langevin dynamics for the spin O(N) model

2025/09/17 by W. T. Ye, Ye, Wenjie, Rongchan Zhu +1
Business, Management and Accounting · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2509.13817

openalex publication_date 2025/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that the large N limit of the Langevin dynamics for the spin O(N) model is given by a mean-field stochastic differential equation (SDE) in both finite and infinite volumes. We establish uniform in N bounds for the dynamics, which enable us to demonstrate convergence to the mean-field SDE with polynomial interactions. Furthermore, the mean-field SDE is shown to be globally well-posed for suitable initial distributions. We also prove the existence of stationary measures for the mean-field SDE. For small inverse temperatures, we characterize the large N limit of the spin O(N) model through stationary coupling. Additionally, we establish the uniqueness of the stationary measure for the mean-field SDE.

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