Networks of reinforced stochastic processes: probability of asymptotic polarization and related general results

In a network of reinforced stochastic processes, for certain values of the parameters, all the agents' inclinations synchronize and converge almost surely toward a certain random variable. The present work aims at clarifying when the agents can asymptotically polarize, i.e. when the common limit inclination can take the extreme values, 0 or 1, with probability zero, strictly positive, or equal to one. Moreover, we present a suitable technique in order to estimate this probability that, along with the theoretical results, has been framed in the general setting of a class of martingales taking values in [0,1].