Coarsening and persistence in the voter model

The voter model is a simple model for coarsening with a nonconserved scalar order parameter. We investigate coarsening and persistence in the voter model by introducing the quantity ${\mathit{P}}_{\mathit{n}}$(t), defined as the fraction of voters who changed their opinion n times up to time t. We show that ${\mathit{P}}_{\mathit{n}}$(t) exhibits scaling behavior that strongly depends on the dimension as well as on the initial opinion concentrations. Exact results are obtained for the average number of opinion changes, 〈n〉, and the autocorrelation function, A(t)\ensuremath{\equiv}\ensuremath{\sum}(-1${)}_{\mathit{n}}^{\mathit{nP}}$\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathit{d}/2}$ in arbitrary dimension d. These exact results are complemented by a mean-field theory, heuristic arguments, and numerical simulations. For dimensions d\ensuremath{\gtrsim}2, the system does not coarsen, and the opinion changes follow a nearly Poissonian distribution, in agreement with mean-field theory. For dimensions d\ensuremath{\le}2, the distribution is given by a different scaling form, which is characterized by nontrivial scaling exponents. For unequal opinion concentrations, an unusual situation occurs where different scaling functions correspond to the majority and the minority, as well as for even and odd n. \textcopyright{} 1996 The American Physical Society.