  In nonlinear voter models the transitions between two states depend in a
nonlinear manner on the frequencies of these states in the neighborhood. We
investigate the role of these nonlinearities on the global outcome of the
dynamics for a homogeneous network where each node is connected to $m=4$
neighbors. The paper unfolds in two directions. We first develop a general
stochastic framework for frequency dependent processes from which we derive the
macroscopic dynamics for key variables, such as global frequencies and
correlations. Explicite expressions for both the mean-field limit and the pair
approximation are obtained. We then apply these equations to determine a phase
diagram in the parameter space that distinguishes between different dynamic
regimes. The pair approximation allows us to identify three regimes for
nonlinear voter models: (i) complete invasion, (ii) random coexistence, and --
most interestingly -- (iii) correlated coexistence. These findings are
contrasted with predictions from the mean-field phase diagram and are confirmed
by extensive computer simulations of the microscopic dynamics.
