Coarsening and persistence in the voter model,
E. Ben-Naim, L. Frachebourg, and P.L. Krapivsky
We investigate coarsening and persistence in the voter model by
introducing the quantity $P_n(t)$, defined as the fraction of voters
who changed their opinion $n$ times up to time $t$. We show that
$P_n(t)$ 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,
$\langle n\rangle$, and the autocorrelation function, $A(t)\equiv \sum
(-1)^nP_n\sim t^{-d/2}$ in arbitrary dimension $d$. These exact
results are complemented by a mean-field theory, heuristic arguments
and numerical simulations. For dimensions $d>2$, the system does not
coarsen, and the opinion changes follow a nearly Poissonian
distribution, in agreement with mean-field theory. For dimensions
$d\neq2$, 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.
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