Large $N$ limit of the Langevin dynamics for the spin $O(N)$ model
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916963185328128 |
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| author | Ye, Wenjie Zhu, Rongchan |
| author_facet | Ye, Wenjie Zhu, Rongchan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_13817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large $N$ limit of the Langevin dynamics for the spin $O(N)$ model Ye, Wenjie Zhu, Rongchan Probability Mathematical Physics 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. |
| title | Large $N$ limit of the Langevin dynamics for the spin $O(N)$ model |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2509.13817 |