Ordering kinetics of the two-dimensional voter model with long-range interactions

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance $r$ with probability $P\left(r\right)\propto r^{-\alpha }$. The model is characterized by different regimes, as $\alpha$ is varied. For $\alpha >4$, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as $L\left(t\right)\propto \sqrt{t}$, until consensus is reached in a time of the order of $NlnN$, with $N$ being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as $\rho \left(t\right)\propto 1/lnt$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For $0<\alpha \le 4$, standard scaling is reinstated and the correlation length increases algebraically as $L\left(t\right)\propto t^{1/z}$, with $1/z=2/\alpha$ for $3<\alpha <4$ and $1/z=2/3$ for $0<\alpha <3$. In addition, for $\alpha \le 3, L\left(t\right)$ depends on $N$ at any time $t>0$. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the $N\to \infty$ limit. In finite systems, consensus is reached in a time of the order of $N$ for any $\alpha <4$.

Figure 1

The points $i,j,k$ and their distances.

Figure 2

$L\left(t\right)$ is plotted against time on double-logarithmic axes, for different values of $\alpha$; see legend. System size is $N={301}^2$. The straight dashed lines are the analytic predictions (16), (56), and (61).

Figure 3

$lnt·C(r,t)$ is plotted against $r/L\left(t\right)$ for $\alpha =5$ at different times (see legend), in a log-log plot. The number of agents is $N={301}^2$. The heavy gray circles are the form (18) with fitting parameters $c_1=0.55,c_2=0,x_0=1.65$. Inset: $C(r,t)$ against $x=r/t^{2/\alpha }$, still with double-logarithmic scales. The green dashed line is the form $x^{-\alpha }$ of Eq. (27), with $x=r/\mathrm{\Lambda }\left(t\right)$.

Figure 4

$C(r,t)$ is plotted against $r$, on a double-logarithmic scale, at different times (see legend), for $\alpha =7/2$. The number of agents is $N={301}^2$. The magenta dashed line is the algebraic behavior (35). Inset: $C(r,t)$ is plotted against $x=r/L\left(t\right)$. The green dashed line is the scaling behavior (47), (48) in the coarsening stage preceding stationarization.

Figure 5

$\mathcal{D}\left(t\right)$ is plotted against $t/N$ (main figure) or against $t/(NlnN)$ (inset). The vertical scale is logarithmic. Different curves correspond to various values of $N$; see legend.

Figure 6

In the left part of the figure, $L\left(t\right)$ is plotted against $t$ for $\alpha =4$ and different system sizes; see legend. In the upper-right part, the same plot is shown, but for $\alpha =5/2$. In the lower-right part, still for $\alpha =5/2$, we plot $N^{-\frac{3-\alpha }{2}}L\left(t\right)$ against $t$.