Synchronization and fluctuations for interacting stochastic systems with individual and collective reinforcement

The Pólya urn is the paradigmatic example of a reinforced stochastic process. It leads to a random (non degenerated) time-limit. The Friedman urn is a natural generalization whose a.s. time-limit is not random anymore. In this work, in the stream of previous recent works, we introduce a new family of (finite) systems of reinforced stochastic processes, interacting through an additional collective reinforcement of mean field type. The two reinforcement rules strengths (one componentwise, one collective) are tuned through (possibly) different rates n −γ. In the case the reinforcement rates are like n −1 , these reinforcements are of Pólya or Friedman type as in urn contexts and may thus lead to limits which may be random or not. Different parameter regimes needs to be considered. We state two kind of results. First, we study the time-asymptotics and show that L 2 and almost sure convergence always holds. Moreover all the components share the same time-limit (so called synchronization). We study the nature of the limit (random/deterministic) according to the parameters' regime considered. Second, we study fluctuations by proving central limit theorems. Scaling coefficients vary according to the regime considered. This gives insights into the different rates of convergence.