Interacting oscillators on random graphs
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One of the most interesting chapter of current research in mathematical physics is focused on interacting particle systems. Mathematical results have been achieved on “simple” models, as the model proposed by Y. Kuramoto in the 70’s and that has become a standard model for describing synchronisation phenomena. In this model each unit is modeled by an oscillator and interacts in the same way with any other unit. This lack of intrinsic topology, geometry and hierarchy is a strong limitation and it has been seriously addressed, in life sciences and physics, notably the generalization of the Kuramoto model on a random graph. The aim of this thesis is to develop a mathematical theory of the Kuramoto model on random graphs and to extend it to more general models: how does the connectivity of the graph influence the dynamics? What is the long-time behaviour of a model with a finite, even if possibly large, number of interacting oscillators? Random graphs, large deviations, mean-field models and statistical mechanics are the key ingredients for this inspiring challenge.