  The influence of zealots on the noisy voter model is studied theoretically
and numerically at the mean-field level. The noisy voter model is a
modification of the voter model that includes a second mechanism for
transitions between states: apart from the original herding processes, voters
may change their states because of an intrinsic, noisy in origin source. By
increasing the importance of the noise with respect to the herding, the system
exhibits a finite-size phase transition from a quasi-consensus state, where
most of the voters share the same opinion, to a one with coexistence. Upon
introducing some zealots, or voters with fixed opinion, the latter scenario may
change significantly. We unveil the new situations by carrying out a systematic
numerical and analytical study of a fully connected network for voters, but
allowing different voters to be directly influenced by different zealots. We
show that this general system is equivalent to a system of voters without
zealots, but with heterogeneous values of their parameters characterizing
herding and noisy dynamics. We find excellent agreement between our analytical
and numerical results. Noise and herding/zealotry acting together in the voter
model yields not a trivial mixture of the scenarios with the two mechanisms
acting alone: it represents a situation where the global-local (noise-herding)
competitions is coupled to a symmetry breaking (zealots). In general, the
zealotry enhances the effective noise of the system, which may destroy the
original quasi--consensus state, and can introduce a bias towards the opinion
of the majority of zealots, hence breaking the symmetry of the system and
giving rise to new phases ...
