LONG TIME DYNAMICS FOR INTERACTING OSCILLATORS ON GRAPHS

• 成果类型: Article
• 署名作者: Coppini, Fabio
• 署名单位: Universite Paris Cite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1680
• 发表日期: 2022
• 页码: 360-391
• 关键词: kuramoto model; Synchronization; inequalities; DIFFUSIONS; networks

摘要

The stochastic Kuramoto model defined on a sequence of graphs is analyzed: the emphasis is posed on the relationship between the mean field limit, the connectivity of the underlying graph and the long time behavior. We give an explicit deterministic condition on the sequence of graphs such that, for any finite time and any initial condition, even dependent on the network, the empirical measure of the system stays close to the solution of the McKean-Vlasov equation associated to the classical mean field limit. Under this condition, we study the long time behavior in the subcritical and in the supercritical regime: in both regimes, the empirical measure stays close to the (possibly degenerate) manifold of stable stationary solutions, up to times which can diverge as fast as the exponential of the size of the system, before large deviation phenomena take over. The condition on the sequence of graphs is derived by means of Grothendieck's inequality and expressed through a concentration in l(infinity) -> l(1) norm. It is shown to be satisfied by a large class of graphs, random and deterministic, provided that the average number of neighbors per site diverges, as the size of the system tends to infinity.