Stationary and dynamic critical behavior of the contact process on the Sierpinski carpet

We investigate the critical behavior of a stochastic lattice model describing a contact process in the Sierpinski carpet with fractal dimension $d=log8/log3$. We determine the threshold of the absorbing phase transition related to the transition between a statistically stationary active and the absorbing states. Finite-size scaling analysis is used to calculate the order parameter, order parameter fluctuations, correlation length, and their critical exponents. We report that all static critical exponents interpolate between the line of the regular Euclidean lattices values and are consistent with the hyperscaling relation. However, a short-time dynamics scaling analysis shows that the dynamical critical exponent $Z$ governing the size dependence of the critical relaxation time is found to be larger then the literature values in Euclidean $d=1$ and $d=2$, suggesting a slower critical relaxation in scale-free lattices.

Figure 1

Configuration at $t=560$ Monte Carlo steps for the contact-process model on a Sierpinski fractal lattice with $L=3^4$ sites and at $p=0.7$. Infected sites are in red (dark gray), noninfected are in white. Black spaces stand for the forbidden sites of the Sierpinski fractal.

Figure 2

Density of infected individuals $\psi$ versus $p$ for distinct $K$ generations in the vicinity of the absorbing state transition.

Figure 3

The moment ratio $m_L\left(p\right)$ as a function of the infection probability $p$ for distinct $K$ generations. The scale invariance at the critical point allowed us to precisely estimate the critical infection probability $p_c=0.682875\left(5\right)$.

Figure 4

Log-log plot of the order parameter versus the linear size $L$ at the critical point. From the best fit to a power law, we estimate the critical exponent ratio $\beta /{\nu }_{\perp }=0.726\left(8\right)$.

Figure 5

The logarithmic derivative of the order parameter versus $L$ at the critical point. From the best fit to a power law, we estimate the critical exponent ${\nu }_{\perp }=0.78\left(1\right)$ in the Sierpinski carpet.

Figure 6

Order-parameter fluctuations $\mathrm{\Delta }\psi (p,K)$ versus $p$ for distinct $K$ generations. The peak in the vicinity of the critical density signalizes the enhancement of the order-parameter fluctuations near the transition.

Figure 7

Log-log plot of the order-parameter fluctuations $\mathrm{\Delta }\psi$ versus $L$ at the critical point. The exponent ratio ${\gamma }^{\prime }/{\nu }_{\perp }=0.42\left(2\right)$ is estimated from the slope of the fitted straight line.

Figure 8

Log-log plot of the order parameter $\psi (K=7)$ versus $p-p_c$ above the critical point. The critical exponent $\beta$ is estimated from the slope of the fitted straight line $\psi \approx {(p-p_c)}^{\beta }$ from which we obtained $\beta =0.576\left(3\right)$.

Figure 9

Log-log plot of the order-parameter fluctuations $\mathrm{\Delta }\psi$ versus $p-p_c$. The critical exponent ${\gamma }^{\prime }$ is estimated from the slope of the fitted straight line $\mathrm{\Delta }\psi \approx {(p-p_c)}^{-{\gamma }^{\prime }}$ from which we obtained ${\gamma }^{\prime }=0.31\left(1\right)$.

Figure 10

Time evolution of the order-parameter at the critical point for $K=8$, averaged over 100 runs. The dynamic scaling law $\psi (p_c,t)\propto t^{-\beta /Z{\nu }_{\perp }}$ holds in the short-time regime. Our best estimate of this critical exponent ratio is reported in Table 2.

Figure 11

Independent determination of the critical point from the short-time dynamics for $K=8$ by using the standard deviation (main frame) and correlation coefficient (inset) method. When plotted against $p$, these quantities reach extremal values at the transition.

Figure 12

Time evolution of the moment ratio $m_L(p_c,t)$ at the critical point. Same conditions as in Fig. 10. The dynamic scaling law $m_L(p_c,t)\propto t^{d/Z}$ holds in the short-time regime. Our estimate of this critical exponent ratio is reported in Table 2.

Figure 13

Time evolution of the order-parameter fluctuations at the critical point. Same conditions as in Fig. 10. The dynamic scaling law $\mathrm{\Delta }\psi (p_c,t)\propto t^{{\gamma }^{\prime }/Z{\nu }_{\perp }}$ holds in the short-time regime. Our estimate of this critical exponent ratio is also reported in Table 2.

Figure 14

Log-Log plot of the derivative of $log\left(\psi (p_c,t)\right)$ near the critical point. Same conditions as in Fig. 10. The dynamic scaling law $dlog\left(\psi (p=p_c,t)\right)/dp\propto t^{1/Z{\nu }_{\perp }}$ holds in the short-time regime. Our best power-law fit provides $Z{\nu }_{\perp }={\nu }_{\parallel }=1.49\left(3\right)$.