Coarsening in the long-range Persistent Voter Model

Abstract

We investigate the coarsening kinetics in a long-range variant of the Persistent Voter Model in space dimension $d=1$ and 2. In this model agents can hold two confidence levels, normal and zealot. If normal, agents take the opinion of others chosen at distance $r$ with probability $P(r) \propto r^{-α}$, with $α>d$. While in the zealot state, agents keep their own opinion. Normal (zealot) agents can become zealots (normal) if their opinion is equal (different) to that of the chosen neighbour. Through numerical simulations we show that, for any values of $α$, the model belongs to the same universality class of the long-range Ising model quenched to a small (non-zero) temperature, similarly to what was already known for the nearest-neighbor case. For the one-dimensional case, we further develop an analytical treatment, which reproduces the $α$-dependence of the correlation length and the functional form of the correlation function. These results not only confirm that the introduction of opinion inertia mitigates the strong interfacial noise present in the voter model, thus reinstating the basic kinetic mechanism of the Ising model, but also expand the applicability of this correspondence.

Figures (6)

• Figure 1: Long-range universality classes, given by the asymptotic value of the exponent $1/z$ in Eq. (\ref{['eq.rhot']}), as a function of the long-range exponent $\alpha>d$, for the VM and the IM quenched to a low (but finite) temperature PhysRevE.49.R27PhysRevE.50.1900PhysRevE.99.011301 for both $d=1$ (top) and $d=2$ (bottom). Notice that in all cases there is a value of $\alpha$, $\alpha_{\mathrm{SR}}$, above which we recover the short-range exponent 1/2 (red regions in the figure).
• Figure 2: Density of interfaces $\rho(t)$ of the PVM in $d=1$ with $L=10^5$ sites for several values of $\alpha$. Averages are over $10^2$ (larger $\alpha$) or $10^3$ (smaller $\alpha$) samples. The results are for the restricted case in which $\theta_i$ is affected by nearest-neighbors only ($r=1$). The straight lines indicate the expected exponent $1/z$ in the correspondent IM universality class (see Fig. \ref{['1suzd1and2']}, top panel).
• Figure 3: Density of interfaces $\rho(t)$ of the long-range PVM in $d=1$ with $\alpha=3/2$ for several system sizes in the unrestricted case. The number of samples ranges from 20 (largest $L$) to 1000 (smallest $L$). There are stronger finite-size effects than in the restricted case and the Ising exponent only becomes clear at late times in the largest system.
• Figure 4: Density of interfaces of the PVM in $d=2$, for different values of $\alpha$ and $L=10^3$, using the fast algorithm where interactions are only considered along the lattice axis. The update of $\theta_i$ is restricted to nearest-neighbors ($r=1$) only and averages are over 100-200 samples. The power-law corresponding to the 2D-IM is well observed for all values of $\alpha$. The down-bending observed at large times is due to finite-size effects.
• Figure 5: The two Laplacians $\Delta_{ij}$ and $\kappa \Delta _{ij}^i$ are plotted against $r=|i-j|$ at different times ($t=100$, 200, 400, 600 and 800 corresponding to decreasing height of the peak) for $\alpha=4$. Inset: the value of $\kappa$, as a function of $\alpha$, that better obeys $\Delta_{ij}\simeq \kappa \Delta _{ij}^i$ for $t=100$. The system size is $N=10^3$ and finite-size effects are still important.