Infinite-randomness critical point in the two-dimensional disordered contact process

We study the nonequilibrium phase transition in the two-dimensional contact process on a randomly diluted lattice by means of large-scale Monte Carlo simulations for times up to ${10}^{10}$ and system sizes up to $8000\times 8000$ sites. Our data provide strong evidence for the transition being controlled by an exotic infinite-randomness critical point with activated (exponential) dynamical scaling. We calculate the critical exponents of the transition and find them to be universal, i.e., independent of disorder strength. The Griffiths region between the clean and the dirty critical points exhibits power-law dynamical scaling 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. Our results are of importance beyond absorbing state transitions because, according to a strong-disorder renormalization group analysis, our transition belongs to the universality class of the two-dimensional random transverse-field Ising model.

Figure 1

(Color online) Phase diagram of the contact process on a site-diluted square lattice (inverse critical infection rate ${\lambda }_c^{-1}$ vs impurity concentration $p$). MCP marks the multicritical point. The black dots show the actual simulation results, the lines are guides to the eye.

Figure 2

(Color online) Time evolution of the average density $\rho \left(t\right)$ for the contact process on an undiluted $(p=0)$ square lattice, starting from a completely active lattice.

Figure 3

(Color online) Time evolution of the survival probability $P_s\left(t\right)$ for impurity concentration $p=0.2$, starting from a single seed site. The clean critical infection rate ${\lambda }_c^0=1.64874$ and the actual critical point ${\lambda }_c=2.10750$ are specially marked.

Figure 4

(Color online) $\ln P_s\left(t\right)$ and $\ln N_s\left(t\right)$ vs $\ln \ln t$ for several infection rates close to the critical point (at least 5000 disorder realizations with 128 trials each, $p=0.2$).

Figure 5

(Color online) $\ln N_s\left(t\right)$ vs $\ln P_s\left(t\right)$ for several infection rates close to the critical point. The straight line is a power-law fit of the asymptotic part of the $\lambda =2.10750$ curve. $p=0.2$, 5000–20 000 disorder realizations with 128 trials each.

Figure 6

(Color online) $\ln \rho \left(t\right)$ vs $\ln \ln (t∕t_0)$ and for several infection rates at and below to the critical point. The dashed line represents $2∕3$ of the critical $\rho$. Inset: Time $t_x$ at which the density falls below $2∕3$ of the critical value as a function of the distance from criticality. The solid line is a power-law fit of $\ln (t_x∕t_0)$ vs $\Delta$. $p=0.2$, 13 disorder realizations with $L=8000$ for $\lambda >2.1$, 250 realizations with $L=2000$ for $\lambda ⩽2.1$.

Figure 7

(Color online) $\ln \rho \left(t\right)$ vs $\ln \Delta \left(t\right)$ at the crossover time $t=t_x$ for the density itself and $\ln P_s\left(t\right)$ vs $\ln \Delta \left(t\right)$ at the crossover time $t=t_x$ for the particle number $N_s$. The dashed lines are power-law fits to the expected behavior $\rho \sim P_s\sim {\mid \Delta \mid }^{\beta }$.

Figure 8

(Color online) $\ln N_s\left(t\right)$ vs $\ln P_s\left(t\right)$ at criticality for impurity concentrations $p=0.1$, 0.2, and 0.3. The dashed lines are power-law fits of the asymptotic parts (18 000 disorder configurations for $p=0.2$, 4000 each for $p=0.1$ and 0.3, 128 spreading trials for each configuration).

Figure 9

(Color online) $\ln \rho \left(t\right)$ vs $\ln t$ for several infection rates in the Griffiths region, ${\lambda }_c^0<\lambda <{\lambda }_c$ ($p=0.2$, 13 disorder realizations with $L=8000$ for $\lambda >2.1$, 250 realizations with $L=2000$ for $\lambda ⩽2.1$). The dashed lines are power-law fits of the long-time behavior to Eq. (24). Inset: Resulting dynamical exponent $z^{\prime }$ as a function of $\lambda$.

Figure 10

(Color online) $\ln G\left(r\right)$ vs $\ln r$ for several infection rates $\lambda$ ($p=0.2$, 13 disorder realizations with $L=8000$). $G\left(r\right)$ is calculated from the configurations at late times of the time evolution $(t⩾4\times {10}^8)$.

Figure 11

(Color online) $\ln \rho \left(t\right)$ vs $\ln \ln \left(t\right)$ for $p=p_c$, $\lambda =5.0$, and $L=8000$. We have simulated three samples; their differences are smaller than the linewidth. The dashed line represents the expected logarithmic long-time decay, $\rho \sim {\left[\ln \left(t\right)\right]}^{-\overline{\delta }}$ with arbitrary prefactor.

Figure 12

(Color online) $\ln P_s\left(t\right)$ and $\ln N_s\left(t\right)$ vs $\ln \ln t$ for the three-dimensional contact process on a diluted lattice ($p=0.3$, $L=400$, at least 5000 disorder realizations with 128 trials each).