Rare regions and Griffiths singularities at a clean critical point: The five-dimensional disordered contact process

We investigate the nonequilibrium phase transition of the disordered contact process in five space dimensions by means of optimal fluctuation theory and Monte Carlo simulations. We find that the critical behavior is of mean-field type, i.e., identical to that of the clean five-dimensional contact process. It is accompanied by off-critical power-law Griffiths singularities whose dynamical exponent $z^{\prime }$ saturates at a finite value as the transition is approached. These findings resolve the apparent contradiction between the Harris criterion, which implies that weak disorder is renormalization-group irrelevant, and the rare-region classification, which predicts unconventional behavior. We confirm and illustrate our theory by large-scale Monte Carlo simulations of systems with up to ${70}^5$ sites. We also relate our results to a recently established general relation between the Harris criterion and Griffiths singularities [Phys. Rev. Lett. 112, 075702 (2014)], and we discuss implications for other phase transitions.

Figure 1

Spreading simulations for the clean five-dimensional contact process: Radius $R$ of the active cloud, survival probability $P_s$, and number of active site $N_s$ vs time $t$ for several infection rates $\lambda$ close to criticality. The system size is ${70}^5$ sites; the data are averages over $2\times {10}^5$ to $6\times {10}^6$ attempts, depending on $\lambda$. The dashed lines are fits of the data for $\lambda =1.1384$ to mean-field critical behavior, $R\sim t^{1/2}$, $P_s\sim t^{-1}$, and $N_s\sim t^0$.

Figure 2

Semilog plot of the decay rate $\epsilon$ vs system volume $L^5$ for several infection rates $\lambda$ slightly above the bulk critical rate ${\lambda }_c^0=1.13845$. The solid lines are fits to $\epsilon ={\epsilon }_{0}exp(-a{L}_{RR}^5)$. (Inset) Decay coefficient $a$ vs distance from bulk criticality $\lambda -{\lambda }_c^0$. The solid line is a power-law fit giving the exponent 1.99(3) and the prefactor $a^{\prime }=0.48\left(6\right)$.

Figure 3

Density of active sites $\rho$ vs time $t$ for a disordered five-dimensional contact process with $p=0.2$ and $c=0.3$. The data are averages over 5 to 100 disorder configurations, depending on $\lambda$, each with ${50}^5$ lattice sites (one run per disorder configuration). The dashed line represents a power-law fit of the critical curve, $\lambda =2.58925\left(20\right)$, giving the exponent $\delta =0.99\left(2\right)$.

Figure 4

Number of active sites $N_s$, survival probability $P_s$, and cloud radius $R$ vs time $t$ for a disordered five-dimensional contact process with $p=0.5$ and $c=0.3$. The data are averages over 1000 to 10 000 disorder configurations of ${60}^5$ sites (1000 trials per disorder configurations). The dashed lines are fits of the data for $\lambda =1.7526$ to mean-field critical behavior, $R\sim t^{1/2}$, $P_s\sim t^{-1}$, and $N_s\sim t^0$.

Figure 5

Density of active sites $\rho$ vs time $t$ for a disordered five-dimensional contact process with $p=0.1$ and $c=0.1$. The data are averages over 100 to 500 disorder configurations, depending on $\lambda$, each with ${51}^5$ lattice sites (one run per disorder configuration). The critical infection rate is $\lambda \approx 5.987$. The dashed line represents the mean-field power law with an arbitrary prefactor. The subcritical curves ($\lambda \le 5.95$) show Griffiths singularities $\rho \sim t^{-d/z^{\prime }}$ rather than exponential decay. (The curve for $\lambda =5.98$ has not reached the asymptotic regime yet.) (Inset) Extrapolation of the Griffiths exponent $d/z^{\prime }$ for $\lambda =5.8,...,5.95$ to criticality (after Ref. [21]).