Nonuniversal and anomalous critical behavior of the contact process near an extended defect

We consider the contact process near an extended surface defect, where the local control parameter deviates from the bulk one by an amount of $\lambda \left(l\right)-\lambda \left(\infty \right)=A{l}^{-s}$, with $l$ being the distance from the surface. We concentrate on the marginal situation $s=1/{\nu }_{\perp }$, where ${\nu }_{\perp }$ is the critical exponent of the spatial correlation length, and study the local critical properties of the one-dimensional model by Monte Carlo simulations. The system exhibits a rich surface critical behavior. For weaker local activation rates $A<A_c$, the phase transition is continuous, having an order-parameter critical exponent, which varies continuously with $A$. For stronger local activation rates $A>A_c$, the phase transition is of mixed order: the surface order parameter is discontinuous; at the same time the temporal correlation length diverges algebraically as the critical point is approached, but with different exponents on the two sides of the transition. The mixed-order transition regime is analogous to that observed recently at a multiple junction and can be explained by the same type of scaling theory.

Figure 1

(a) Time dependence of the survival probability measured in numerical simulations for different values of $A$, $A=0.5$, 1, 1.5, 2, 2.5, 3, 3.25, 3.5, 4, 4.5, 5, 5.5, 6, 7, 8 (from bottom to top) in the bulk critical point $\lambda ={\lambda }_c$. The data at the estimated tricritical point $A_c=3.25$ are shown by circles. (b) Effective decay exponents for the same data (from top to bottom).

Figure 2

Numerical estimates of various surface critical exponents plotted against $A$. The curves fit to the data points (quadratic function for $\delta$ and linear for ${\delta }^{\prime }$ and ${\nu }_{\parallel ,1}$) seem to cross the $x$ axis at $A=A_c\approx 3.25$.

Figure 3

Time dependence of the quantity $tdP\left(t\right)/dt$ in a log-log plot for different values of $A$ in the discontinuous regime, $A=3.25$, 3.5, 4, 4.5, 5, 5.5, 6, 7, 8 (from top to bottom). The asymptotic slope of the curves is $-{\delta }^{\prime }$ according to Eq. (5).

Figure 4

Numerically estimated surface order parameters as a function of $A$. The full (red) line represents the extrapolated behavior according to Eq. (6). The slope of the straight line in the inset is 0.40.

Figure 5

The logarithm of the survival probability as a function of $ln(lnt)$ for different values of $A$. The slope of the straight line is $-0.53$.

Figure 6

The inverse of the square of the survival probability plotted against $lnt$ for different values of $A$.

Figure 7

Scaling plot of the survival probability in the inactive phase for $A=4$. The parameter ${\nu }_{\parallel ,1}=2.0$ was used. The inset shows the unscaled data.

Figure 8

Scaling plot of the derivative of the survival probability in the active phase for $A=5$. The parameters ${\delta }^{\prime }=0.37$ and ${\nu }_{\parallel }^{\prime }=1.7338$ were used. The inset shows the unscaled data.