Time-Delay Induced Dimensional Crossover in the Voter Model

We investigate the ordering dynamics of the voter model with time-delayed interactions. The dynamical process in the $d$-dimensional lattice is shown to be equivalent to the first passage problem of a random walker in the ($d+1$)-dimensional strip of a finite width determined by the delay time. The equivalence reveals that the time delay leads to the dimensional crossover from the ($d+1$)-dimensional scaling behavior at a short time to the $d$-dimensional scaling behavior at a long time. The scaling property in both regimes and the crossover time scale are obtained analytically, which are confirmed with the numerical simulation results.

Figure 1

Domain wall density $\rho$ for the DVM with fixed-$\tau$ distribution, with ${\tau }_0=2,\dots ,32$. The symbols are obtained from the discrete time Monte Carlo simulations while the solid lines are from the continuous time SC simulations. Also shown with the dashed line is the Monte Carlo data for ${\rho }_2$ at ${\tau }_0=16$.

Figure 2

Diagrams for the DVM in one dimension. (a) Space-time diagram representation of the DVM dynamics. The active and inactive periods of $W_4$ are also drawn. (b) Typical motion of $z$ and $m$ variables defined in the text. The filled (open) circles represent the positions where two walkers $W_i$ and $W_j$, with even (odd) $x(0)=x_i(0)-x_j(0)$, can coagulate. (c) Diffusion of the $R$ walker in the two dimensional $q=(x,z)$ plane of strip geometry with the target $D$.

Figure 3

Scaling plot of $\rho (t)$ obtained from the SC simulations against $t/[{\tau }_0^3(\ln {\tau }_0{)}^2]$. The inset shows that $t^{*}/{\tau }_0^3$, where $\rho (t^{*})=0.05(\times )$ and 0.025 (square), is linear in $(\ln {\tau }_0{)}^2$, which confirms the scaling $T({\tau }_0)=(1+{\tau }_0{)}^3[\ln (1+{\tau }_0){]}^2$.

Figure 4

Log-linear plots of ${\rho }_2(t)$ against $t/{\tau }_0$ for several values of ${\tau }_0$ obtained from the SC simulations. The linear segment confirms the logarithmic scaling in the short time region. The dashed straight line is a guide to an eye. The scaling plot in the inset confirms the scaling form in (10) in the $t\gg {\tau }_0^3$ limit.