Universal and nonuniversal features of the generalized voter class for ordering dynamics in two dimensions

By considering three different spin models belonging to the generalized voter class for ordering dynamics in two dimensions [Dornic et al., Phys. Rev. Lett. 87, 045701 (2001)], we show that they behave differently from the linear voter model when the initial configuration is an unbalanced mixture of up and down spins. In particular, we show that for nonlinear voter models the exit probability (probability to end with all spins up when starting with an initial fraction $x$ of them) assumes a nontrivial shape. This is the first time a nontrivial exit probability is observed in two-dimensional systems. The change is traced back to the strong nonconservation of the average magnetization during the early stages of dynamics. Also the time needed to reach the final consensus state $T_N\left(x\right)$ has an anomalous nonuniversal dependence on $x$.

Figure 1

Main: Finite-size scaling of the exit probability at $x=0.25$ as a function of system size in the NLV model in $d=2$, computed over at least $5\times {10}^5$ independent realizations of the dynamics for each system size. Inset: Susceptibility and correlation length as a function of $\Delta =|{\varepsilon }_c-\varepsilon |$ on lattices of size $L=100$ and $L=5000$, respectively.

Figure 2

(a) Exit probability as a function of the initial density for different system sizes in the NLV model and the $q=4$ qV model at the respective GV critical points. (b) Normalized consensus time as a function of $x$ for the same models.

Figure 3

Average magnetization $\varphi \left(t\right)$ as a function of time for different system sizes in the NLV model and the $q=4$ qV model at the respective GV critical points, for initial conditions with magnetization $\varphi \left(0\right)=-0.5$, corresponding to $x=0.25$.

Figure 4

Average drift as a function of the magnetization for different system sizes in the NLV model in $d=2$ at the critical point, for system starting at an initial magnetization $\varphi \left(0\right)=-0.5$ ($x=0.25$). Top: $t<100$. Bottom: $t>100$.

Figure 5

Main: Finite-size scaling of the exit probability at $x=0.25$ as a function of system size in the $q=4$ qV model in $d=2$, computed over at least $5\times {10}^5$ independent realizations of the dynamics for each system size. Inset: Susceptibility and correlation length as a function of $\Delta =|{\varepsilon }_c-\varepsilon |$ for the $q=4$ qV model on lattices of size $L=100$ and $L=5000$, respectively.

Figure 6

Average drift as a function of the magnetization for different system sizes in the $q=4$ voter model in $d=2$ at the critical point, for a system starting at an initial magnetization $\varphi \left(0\right)=-0.5$ ($x=0.25$). Top: $t<50$. Bottom: $t>50$.

Figure 7

Main plot: Consensus time as a function of $x$ for different system sizes in the KKE model at the delocalization transition in $d=2$. Inset: Evolution of $T_N\left(x\right)/T_N\left(0.5\right)$ for $x=0.25$ as a function of the system size $L$.