Dynamic phase transition in the contact process with spatial disorder: Griffiths phase and complex persistence exponents

We present a model which displays the Griffiths phase, i.e., algebraic decay of density with continuously varying exponents in the absorbing phase. In the active phase, the memory of initial conditions is lost with continuously varying complex exponents in this model. This is a one-dimensional model where a fraction $r$ of sites obey rules leading to the directed percolation class and the rest evolve according to rules leading to the compact directed percolation class. For infection probability $p\le p_c$, the fraction of active sites $\rho \left(t\right)=0$ asymptotically. For $p>p_c, \rho \left(\infty \right)>0$. At $p=p_c, \rho \left(t\right)$, the survival probability from a single seed and the average number of active sites starting from single seed decay logarithmically. The local persistence $P_l\left(\infty \right)>0$ for $p\le p_c$ and $P_l\left(\infty \right)=0$ for $p>p_c$. For $p\ge p_s$, local persistence $P_l\left(t\right)$ decays as a power law with continuously varying exponents. The persistence exponent is clearly complex as $p\to 1$. The complex exponent implies logarithmic periodic oscillations in persistence. The wavelength and the amplitude of the logarithmic periodic oscillations increase with $p$. We note that the underlying lattice or disorder does not have a self-similar structure.

Figure 1

(a) Overview of time evolution of average density $\rho \left(t\right)$ of a 1D system of size $N=2.5\times {10}^6$ for $r=0.5$ and $p\ge p_c$ in the range 0.64 to 0.85 (from bottom to top), where $p_c=0.651$. We find that $\rho \left(t\right)\sim t^{-\delta }$ for $p$ is close to $p_c$ and $p<p_c$. (b) Log-log plot of $\rho \left(t\right)\times t^{\delta }\text{vs}$ time in the Griffiths region for a system with $p<p_c$ in the range 0.61 to 0.648 (from top to bottom). Exponent $\delta$ changes continuously and reaches 0 as $p\to p_c$.

Figure 2

The time evolution of density of active sites $\rho \left(t\right)$ for a system of size $2.5\times {10}^6$ with $p<p_c, p=p_c$, and $p>p_c$. $\rho \left(t\right)$ decays as ${[ln(t/t_0)]}^{-\overline{\delta }}$ where $\overline{\delta }=0.381$ only at $p=p_c$.

Figure 3

(a) The time evolution of the survival probability $P_s\left(t\right)$ for disorder concentration $r=0.5$ starting with a single seed, for $p<p_c, p=p_c$, and $p>p_c$. $P_s\left(t\right)$ decays as ${[ln(t/t_0)]}^{-\overline{\delta }}$ with $\overline{\delta }=0.381$. (b) The time evolution of average number of active sites in a cluster starting with a single seed $N\left(t\right)$ for $p>p_c,p=p_c$, and $p<p_c$. $N\left(t\right)$ decays as ${[ln(t/t_0)]}^{\mathrm{\Theta }}$ with $\mathrm{\Theta }=1.236$ and $t_0=0.2$.

Figure 4

Time evolution of local persistence $P_l\left(t\right)$ for a system of size $2.5\times {10}^6$ with $r=0.5$ and $p<p_s$ in the range 0.62 to 0.654 (from top to bottom).

Figure 5

Scaling plot of the local persistence $P_l\left(t\right)$ at $r=0.5$. with $p=p_s=0.655$ for different lattice sizes $N$. The time required for saturation scales as $N^z$ and the saturation value of $P_l\left(t\right)$ scales with $N^{-z\theta }$ where $N=80,160,320,640,1280$ (from bottom to top). Inset: Time evolution of $P_l\left(t\right)$ with $r=0.5, p=p_s$ for various sizes of lattice $N=80,160,320,640,1280$ (from top to bottom).

Figure 6

(a) Time evolution of $P_l\left(t\right)$ with $r=0.5$ and $p\ge p_s$. Power law is obtained at $p\ge p_s$ in the range 0.655 to 0.99 (from bottom to top). Clearly the persistence exponent $\theta$ changes continuously in this range. (b) Log-log plot of $P_l\left(t\right)\times t^{\theta }$ with $r=0.5$ and $p>p_s,$ in the range 0.88 to 0.99 (from bottom to top) $p\to 1$. The log-periodic oscillations are clearly evident in this figure. The $y$ axis is multiplied by arbitrary constants for better visualization.

Figure 7

(a) Starting with ${10}^4$ configurations, we plot the number of surviving configurations $\text{vs}t/T_k$ for CDP clusters of $k$ sites surrounded by inactive sites for $p=0.96$. We consider $k=3–8$. $T_k$ is the average time taken by a cluster of $k$ CDP sites to become inactive. Clearly, the distribution of lifetimes is different for odd (at the bottom) and even (at the top) values of $k$. (b) We plot $T_k\text{vs}k$. Ignoring odd-even oscillations, it can be fitted as $T_k\propto exp\left({\gamma }_{p}k\right)$ where ${\gamma }_p=0.915,1.1,1.6$ for $p=0.85,0.9,0.96$. (c) We plot the relation $T_k/exp\left({\gamma }_{p}k\right)\text{vs}k$. The odd-even oscillations are evident.

Figure 8

Semilogarithmic plot of $\text{Re}\left({\theta }_l\right)=\theta$ and $\text{Im}\left({\theta }_l\right)/2\pi ={\theta }^{\prime }/2\pi$ for $p=0.96$ and varying disorder fraction $r$.