Synchronization of mean-field models on the circle
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918107838152704 |
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| author | Polyanskiy, Yury Rigollet, Philippe Yao, Andrew |
| author_facet | Polyanskiy, Yury Rigollet, Philippe Yao, Andrew |
| contents | This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $β\ge -0.16$, which significantly extends the previous bound of $0\le β\le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $β< -2/3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Synchronization of mean-field models on the circle Polyanskiy, Yury Rigollet, Philippe Yao, Andrew Dynamical Systems Machine Learning Analysis of PDEs Optimization and Control This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $β\ge -0.16$, which significantly extends the previous bound of $0\le β\le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $β< -2/3$. |
| title | Synchronization of mean-field models on the circle |
| topic | Dynamical Systems Machine Learning Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2507.22857 |