Percolation and coarsening in the bidimensional voter model

We study the bidimensional voter model on a square lattice with numerical simulations. We demonstrate that the evolution takes place in two distinct dynamic regimes; a first approach towards critical site percolation and a further approach towards full consensus. We calculate the time dependence of the two growing lengths, finding that they are both algebraic but with different exponents (apart from possible logarithmic corrections). We analyze the morphology and statistics of clusters of voters with the same opinion. We compare these results to the ones for curvature driven two-dimensional coarsening.

Figure 1

Snapshots of the voter (first row) and Ising (second and third rows) models on a $2d$ square lattice with linear size $L=640$ and periodic boundary conditions. The Ising model has been quenched to $T=0$ (second row) and $T_c$ (third row). The images were taken at the times indicated below the snapshots.

Figure 2

Snapshots of the voter model with linear size $L=640$ and periodic boundary conditions. The times shown are $t_i=2^i$, with $i=12$–19. With the same convention as in Fig. 1, the $+1$ voters are in medium gray (red), while the $-1$ voters are in white. We highlight here the percolating clusters with different colors. Percolating clusters of $+1$ opinion are clear gray (green) and of $-1$ opinion are black (blue). A single domain of “$+1$” voters was reached at $t\approx 6.7\times {10}^5$.

Figure 3

Concentration of active interfaces $\rho \left(t\right)=[1-\langle s_{\mathit{x}}\left(t\right)s_{\mathit{x}+{\mathrm{e}}_i}\left(t\right)\rangle ]/2$ as a function of time. Data for $L=160$ and 640 are presented in a log-linear form. The error bars are smaller than the symbol sizes. The dotted (black) line represents the analytic asymptotic form $\rho \left(t\right)=a/(2lnt+b)$ with $a=3.14$ and $b=5.54$. In the inset we show $b=\pi /\rho \left(t\right)-2lnt$ as a function of $t$ for different lattice sizes together with the analytic value $5.54$ shown with a horizontal dashed line. See the main text for a discussion.

Figure 4

(a) The average consensus time $\langle \mathcal{T}\rangle$ (red diamonds) for different values of the lattice linear size $L$ in a log-log plot. The value of $\langle \mathcal{T}\rangle$ has been computed over approximately ${10}^5$ samples for the smaller sizes and over about $7\times {10}^3$ samples for $L=640$. We report also the approximate width of the distribution, computed as the standard deviation of the data collected, $\delta \mathcal{T}={\left[N_s^{-1}{\sum }_{i=1}^{N_s}{\left(t_i-\langle \mathcal{T}\rangle \right)}^2\right]}^{1/2}$, as vertical dashes with length $2\delta \mathcal{T}$ on each data point. Roughly, $\delta \mathcal{T}/\langle \mathcal{T}\rangle \simeq 0.74$ for all the sizes. The function $C{x}^{\eta }$ was fitted to the few data points available, yielding $\eta \simeq 2.21$ and $C\simeq 0.75$. By introducing a logarithmic prefactor, $C{x}^{\eta }lnx$, the fit yields instead $\eta \simeq 2.05$ and $C\simeq 0.33$. In the upper part of the same panel we report the skewness $\gamma$ and the kurtosis $\kappa$ of the sampled distributions in order to quantify the deviations from a Gaussian form. (b) The distribution of consensus times for different system sizes $L$ given in the key.

Figure 5

Rescaled histogram of the consensus time $p_{\mathrm{CT}}(\mathcal{T},L)$ for different lattice sizes. Time $\mathcal{T}$ is rescaled by $L^{2}lnL$, while the value of $p_{\mathrm{CT}}(\mathcal{T},L)$ is multiplied by $L^{\alpha }$. The best collapse of the different curves was obtained for $\alpha \simeq 2.22$.

Figure 6

(a) The logarithm of the persistence probability, $ln{P}_0$, against ${ln}^{2}t$, for a system of size $L=640$. A linear fit to the data in this scale, $f\left(x\right)=b-ax$, gives $a\simeq 0.26$ and $b\simeq 0.36$ (the fitting curve is showed as a dashed blue line). The persistence has been calculated as an average over $5\times {10}^4$ realizations of the dynamics. Data for $L=160$ are shown in the inset along with their fitting curve. (b) Time dependence of the autocorrelation with the initial configuration, $A_0$, in a double logarithmic scale. The function $f\left(t\right)=a{t}^{-b}$ has been fitted to the numerical data, yielding $b=0.995$ and $a=0.308$, which are both in agreement with what theory predicts, i.e., $A\left(t\right)\sim {\left(\pi t\right)}^{-1}$, shown with a dark dashed segment. In the inset we show data for $L=160$ analyzed in a similar fashion.

Figure 7

The two-time autocorrelation function $A(t,t_0)$ against time-delay $t-t_0$ for different values of the waiting-time $t_0$. Panels (a) and (b) show data for systems with linear size $L=160$ and $L=640$, respectively.

Figure 8

The two-time correlation function $A(t,t_0)$ times $ln{t}_0$ against $(t/t_0-1)$, for different values of the waiting time $t_0$, in a log-log scale. A segment with slope $-1$ is shown as a guide-to-the-eye to stress the good agreement with the exponent found analytically.

Figure 9

Average number of wrapping domains $N_P(t,L)$ against rescaled time $t/L^{{\alpha }_P}$, with ${\alpha }_P=1.667$, for systems of sizes $L=20,40,80,160$. As $t\to +\infty$ the curves should converge to 1 (single domain state).

Figure 10

The largest cluster. The horizontal axes are $t/L^2$ and $t/L^{{\alpha }_P}$ with ${\alpha }_P=1.667$ in all panels in the left and right columns, respectively. [(a) and (b)] The averaged number of external interfaces (only for nonpercolating clusters). Its area $A_c$ normalized by $L^2$ in (c) and $L^{D_A}$ with $D_A=1.89583$ in (d). [(e) and (f)] Its total perimeter length in the voter and $2d\mathrm{IM}$ quenched to $T=0$ (inset) normalized by $L^{D_H}$ with $D_H=1.75$.

Figure 11

Instantaneous domain area number densities. Main plot: $2d$ voter model with linear size $L=640$. Inset: $2d\mathrm{IM}$ quenched to $T=0$ from $T_0\to \infty$ and $L=640$. In both cases the curves are presented in a double logarithmic scale and the times at which the data are collected are given in the key.

Figure 12

Time-dependent number density of domain areas, $n_d$, multiplied by $A^{\nu }$ with $\nu \simeq 1.98$ for $L=640$. As the system comes closer to the percolating state, the number density of domain areas tends to the time-independent form $n_d(A,t)\sim c_d{A}^{\nu }$, save for a correction due to wrapping domains in the bump. The value of $\nu$ was estimated by fitting the aforementioned functional form to the data corresponding to the latest time. The horizontal line is at $c_d\approx 0.022$.

Figure 13

Number density of domain areas multiplied by $A^{\nu }$ against the rescaled area $A/t^{\alpha }$ for $L=640$. The exponent $\nu$ takes the same values as in Fig. 12: $\nu \simeq 1.98$. The value of $\alpha$ that yields the best time scaling is $\alpha \simeq 1.19$. A fit to the function $f\left(u\right)=c{u}^a$ of the data at $t=64$ over the interval $[1,100]$ yields $a\approx 0.26$. The red dashed line was added to better visualise this power-law behavior. Inset: The same scaling plot for the $2d\mathrm{IM}$ after a quench to $T=0$, at five short times given in the key. The plateau is at $2c_d\simeq 0.044$ as explained in Ref. [21]. The power law shown with a dashed line was drawn with the same value of the exponent $a$ as for the voter model.

Figure 14

(a) Log-linear plot of the space-time correlation function along a principal direction of the lattice, i.e., $C(x,t)=\langle s_{x{\mathrm{e}}_i}\left(t\right)s_{\mathit{0}}\left(t\right)\rangle$, at different times for a system with linear size $L=640$. (b) The same data as in panel (a) multiplied by $lnt$ and plotted against the scaled distance $x/\sqrt{t}$ in linear-log scale. See the main text for a discussion.

Figure 15

$C(x,t)lnt$ against scaled distance $x/t^{1/z_d}$ with $z_d=2$ the dynamical exponent in a linear-log plot. In panel (a) data are for $L=80$ at different times given in the key. Panel (b) reports data for different $L_i$ and the corresponding times $t_i$ such that $L_i^{{\alpha }_P}/t_i$ is held constant. ${\alpha }_P\approx 1.667$ is the exponent in the relation $t_P\sim L^{{\alpha }_P}$, with $t_P$ the time needed for stable percolating domains to establish (see Sec. 3f for more details on $t_P$).