Critical behavior and Griffiths effects in the disordered contact process

We study the nonequilibrium phase transition in the one-dimensional contact process with quenched spatial disorder by means of large-scale Monte Carlo simulations for times up to ${10}^9$ and system sizes up to ${10}^7$ sites. In agreement with recent predictions of an infinite-randomness fixed point, our simulations demonstrate activated (exponential) dynamical scaling at the critical point. The critical behavior turns out to be universal, even for weak disorder. However, the approach to this asymptotic behavior is extremely slow, with crossover times of the order of ${10}^4$ or larger. In the Griffiths region between the clean and the dirty critical points, we find power-law dynamical behavior with continuously varying exponents. We discuss the generality of our findings and relate them to a broader theory of rare region effects at phase transitions with quenched disorder.

Figure 1

Phase diagrams of the disordered contact process. Left: Birth rate $\lambda$ vs impurity concentration $p$ for fixed impurity strength $c=0.2$. Right: $\lambda$ vs $c$ for fixed $p=0.3$.

Figure 2

Overview of the time evolution of the density for a system of ${10}^6$ sites with $p=0.3$ and $c=0.2$. The clean critical point ${\lambda }_c^0\approx 3.298$ and the dirty critical point ${\lambda }_c\approx 5.24$ are specially marked.

Figure 3

Time evolution of the density at the clean critical point ${\lambda }_c^0=3.298$ for systems of ${10}^7$ sites with $c=0.2$ and several $p$. The straight lines are fits to the stretched exponential $\ln \rho \left(t\right)\sim -E{t}^{d∕(d+z)}$ predicted in Eq. (22) with $d=1$ and the clean $z=1.580$. Inset: Decay constant $E$ vs ${\tilde{p}}^{z∕(d+z)}$.

Figure 4

Log-log plot of the density time evolution in the Griffiths region for systems with $p=0.3,c=0.2$, and several birth rates $\lambda$. The system sizes are ${10}^7$ sites for $\lambda =3.5,3.7$, and ${10}^6$ sites for the other $\lambda$ values. The straight lines are fits to the power law $\rho \left(t\right)\sim t^{-1∕z^{\prime }}$ predicted in Eq. (24). Inset: dynamical exponent $z^{\prime }$ vs birth rate $\lambda$.

Figure 5

${\rho }^{-1∕\overline{\delta }}$ vs $\ln \left(t\right)$ for a system of ${10}^4$ sites with $p=0.3$ and $c=0.2$. The filled circles mark the critical birth rate ${\lambda }_c=5.24$, and the straight line is a fit of the long-time behavior to Eq. (18).

Figure 6

Time evolution of the density for systems of ${10}^4$ sites with $p=0.3$ and $c=0.2$, 0.4, 0.6, and 0.8 at their respective critical points plotted as ${\rho }^{-1∕\overline{\delta }}$ vs $\ln \left(t\right)$ as in Fig. 5.

Figure 7

Replot of the data of Fig. 6 in log-log form. The dashed line shows a power-law decay with the clean critical exponent $\delta =0.159$.

Figure 8

Survival probability $P_s$ and particle number $N$ in the active cluster for a simulation starting from a single active site. The disorder parameters are $p=0.3$ and $c=0.2$.

Figure 9

Time evolution of the density in the active phase for a system with $p=0.3$ and $c=0.2$. The system sizes are ${10}^4$ for $\lambda ⩾5.8$ and ${10}^3$ otherwise. Inset: Steady state density vs distance from critical point. The solid line is a fit of the data for $\lambda -{\lambda }_c<0.3$ to the expected dirty power law ${\rho }_{\mathrm{stat}}\sim {(\lambda -{\lambda }_c)}^{\beta }$ with $\beta =0.38197$. The dashed line is a fit of the data for $\lambda -{\lambda }_c>0.3$ to a power law with the clean $\beta =0.2765$.