Contact process in heterogeneous and weakly disordered systems

The critical behavior of the contact process (CP) in heterogeneous periodic and weakly disordered environments is investigated using the supercritical series expansion and Monte Carlo (MC) simulations. Phase-separation lines and critical exponents $\beta$ (from series expansion) and $\eta$ (from MC simulations) are calculated. A general analytical expression for the locus of critical points is suggested for the weak-disorder limit and confirmed by the series expansion analysis and the MC simulations. Our results for the critical exponents show that the CP in heterogeneous environments remains in the directed percolation universality class, while for environments with quenched disorder, the data are compatible with the scenario of continuously changing critical exponents.

Figure 1

(Color online) Periodic 1D lattices $AB$ (엯), $AAB$ (◇), and $AABB$ (+): (a) critical points obtained by series expansions in comparison with analytical prediction for the critical line ${\mu }_c=\left({\mu }_A^{1-q}{\mu }_B^q\right)$, $q=1∕2$ (—) and $q=1∕3$ (- - -) for ${\mu }_c=0.303228$ [14]; (b) critical exponent $\beta$ in comparison with series expansion value $\beta =0.2769$ (- - -) [14] for the homogeneous case.

Figure 2

(Color online) The phase diagram (a) and scaling exponent $\beta \left({\mu }_A\right)$, (b) and inset of (a), for disordered lattices with various degrees of disorder: $q=0.04$ (엯), 0.3 (▵), and 0.5 (◻) obtained by series expansion. The lines represent the theoretical prediction according to Eq. (1), ${\mu }_c=\left({\mu }_A^{1-q}{\mu }_B^q\right)$, for $q=0.04$ (—), 0.3 (∙∙ - ∙∙), and 0.5 (∙∙∙) with ${\mu }_c=0.303228$ [14]. The triangle in (a) marks the region for which the MC simulations shown in Fig. 3 have been run. The dashed lines in (b) and the inset of (a) show the value of $\beta$ for the homogeneous case, ${\beta }_c=0.2769$ (- - -) [14]. The arrow in (b) marks the homogeneous critical point.

Figure 3

(Color online) Number of infected sites vs time obtained by MC simulations of the disordered CP with $q=0.5$ and ${\mu }_A=0.25$, averaged over 1500 runs and at least 30 disorder realizations. The main figure is in double-logarithmic scale according to the conventional scaling while the inset demonstrates the same data in the activated scaling picture [12]. The curves from top to bottom correspond to 0.3628 (▵), 0.3680 (◻), and 0.3728 (◇).