Coarsening in the Persistent Voter Model: analytical results

We investigate the coarsening dynamics of a simplified version of the persistent voter model in which an agent can become a zealot – i.e. resistent to change opinion – at each step, based on interactions with its nearest neighbors. We show that such a model captures the main features of the original, non-Markovian, persistent voter model. We derive the governing equations for the one-point and two-point correlation functions. As these equations do not form a closed set, we employ approximate closure schemes, whose validity was confirmed through numerical simulations. Analytical solutions to these equations are obtained and well agree with the numerical results.

💡 Research Summary

The paper introduces a simplified, Markovian version of the Persistent Voter Model (PVM) that retains the essential physics of the original non‑Markovian formulation while allowing analytical treatment. In the classic PVM each agent possesses a confidence variable ηi that increases through repeated interactions with like‑minded neighbors; once ηi exceeds a threshold the agent becomes a zealot, i.e., it no longer changes its opinion. The authors replace the continuous confidence by a binary zealot variable θi (θi=+1 for zealot, –1 for normal voter) and set the confidence increment Δη=1, which makes the dynamics fully Markovian.

The state of the system is described by spins Si=±1 (opinions) and zealot variables θi. Transition rates for opinion flips and zealot updates are given by equations (1) and (2), which depend on the local configuration of neighboring spins and zealots. From the master equation they derive evolution equations for arbitrary correlation functions ⟨α1α2…αm⟩, where each α can be Si or θi. Specialising to one‑ and two‑point correlators yields a set of coupled differential equations (5)–(7). These equations are not closed because higher‑order correlators such as C_{i,i+2δ} appear.

To close the hierarchy the authors adopt two approximations. First, they assume statistical independence between spins and zealot variables (C_{ij}=Ci Cj, etc.). Under this assumption the magnetization C_i is conserved, consistent with random initial conditions. Second, they introduce an ansatz for the next‑nearest‑neighbour correlator: C_{i,i+2δ}≈C_{q,i,i+δ}, where q is a dimension‑dependent constant. Numerical simulations show that q=2 in one dimension and q=4/3 in two dimensions give an excellent fit.

Using these approximations they define the density of active sites ρ=(1−C_{i,i+δ})/2 and the fraction of non‑zealots φ=(1−C_i)/2. The closed dynamical system becomes

 dφ/dt = ρ − φ,  dρ/dt = φ² d