Contact process in disordered and periodic binary two-dimensional lattices

The critical behavior of the contact process (CP) in disordered and periodic binary two-dimensional (2D) lattices is investigated numerically by means of Monte Carlo simulations as well as via an analytical approximation and standard mean field theory. Phase-separation lines calculated numerically are found to agree well with analytical predictions around the homogeneous point. For the disordered case, values of static scaling exponents obtained via quasistationary simulations are found to change with disorder strength. In particular, the finite-size scaling exponent of the density of infected sites approaches a value consistent with the existence of an infinite-randomness fixed point as conjectured before for the $2D$ disordered CP. At the same time, both dynamical and static scaling exponents are found to coincide with the values established for the homogeneous case thus confirming that the contact process in a heterogeneous environment belongs to the directed percolation universality class.

Figure 1

(Color online) The phase diagram for the CP on a disordered lattice with sites of recovery rates ${\epsilon }_A$ or ${\epsilon }_B$ drawn from the distribution Eq. (5) for the case $x=0.5$ obtained from MC simulation (dots), mean field (upper dashed line), and the analytical expression Eq. (8) (solid line). Inset shows the deviation $\Delta \left(d\right)$ as defined in the text for both mean field (◻, blue) and Eq. (8) (○, red).

Figure 2

(Color online) Top panel: The terms $B_m$ as defined in Eq. (17) for a linear $AB$ chain of size $N=4$ (left black peak, ◇) and $N=6$ (right red peak, ◻) normalized by their maxima. Bottom panel: The correction terms $C_k$ as defined in Eq. (19) as a function of the relative difference between powers $k∕m_{\max }$, where $m_{\max }$ is the location of the respective maximum in the upper panel. Symbols as before, all values have been normalized to the corresponding homogeneous contributions $C_0$.

Figure 3

(Color online) Illustration of the data used for extraction of the exponent $x$ for the case ${ϵ}_a=0.5$. Straight lines are least-squares fits to the subsets of system sizes indicated in the legend and the top dashed line is the prediction for the exponent value at an infinite-randomness fixed point. The resulting slopes corresponding to different subsets are found to differ by less than 2%.

Figure 4

(Color online) (a)–(e) The phase diagram for the CP on the lattices (i)–(v) as defined in the text. The circles represent the MC data, the solid line is given by Eq. (23) for the specific lattice, and the dashed line corresponds to the mean field result taken from Table . The inset shows the deviation $\Delta \left(d\right)$ for both approximations [mean $\text{field}=◻$, Eq. $\left(23\right)=○$]. (f) The deviation $\Delta$ as defined in the text at criticality for ${ϵ}_A=0.9$ for the CP on a lattice such as case (ii) but with variable linear size $L$ of the square contiguous regions of $A$ or $B$ sites.