{"ID":2888886,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.22857","arxiv_id":"2507.22857","title":"Synchronization of mean-field models on the circle","abstract":"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 $β\u003c -2/3$.","short_abstract":"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 gener...","url_abs":"https://arxiv.org/abs/2507.22857","url_pdf":"https://arxiv.org/pdf/2507.22857v1","authors":"[\"Yury Polyanskiy\",\"Philippe Rigollet\",\"Andrew Yao\"]","published":"2025-07-30T17:31:57Z","proceeding":"math.DS","tasks":"[\"math.DS\",\"cs.LG\",\"math.AP\",\"math.OC\"]","methods":"[\"Transformer\"]","has_code":false}