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We investigate the long-time asymptotics of the fluctuation SPDE in theKuramoto synchronization model. We establish the linear behavior for large timeand weak disorder of the quenched limit fluctuations of the empirical measureof the particles around its McKean-Vlasov limit. This is carried out through aspectral analysis of the underlying unbounded evolution operator, usingarguments of perturbation of self-adjoint operators and analytic semigroups. Westate in particular a Jordan decomposition of the evolution operator which isthe key point in order to show that the fluctuations of the disordered Kuramotomodel are not self-averaging.

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model