Supercritical series expansion for the contact process in heterogeneous and disordered environments

The supercritical series expansion of the survival probability for the one-dimensional contact process in heterogeneous and disordered lattices is used for the evaluation of the loci of critical points and critical exponents $\beta$. The heterogeneity and disorder are modeled by considering binary regular and irregular lattices of nodes characterized by different recovery rates and identical transmission rates. Two analytical approaches based on nested Padé approximants and partial differential approximants were used in the case of expansions with respect to two variables (two recovery rates) for the evaluation of the critical values and critical exponents. The critical exponents in heterogeneous systems are very close to those for the homogeneous contact process thus confirming that the contact process in periodic heterogeneous environment belongs to the directed percolation universality class. The disordered systems, in contrast, seem to have continuously varying critical exponents.

Figure 1

(Color online) Comparison of critical points (a) and critical exponents (b) of the $AB$ system obtained from series expansion for different orders $N$: $N=10$ $(엯)$, $N=17$ $(◇)$, and $N=24$ $(+)$. The symmetric transformation T1 was used. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ are given by Eq. (31) in (a) and by $\beta$ at the homogeneous critical point, $\beta =0.2769$, from series expansions [14] for the CP, in (b).

Figure 2

(Color online) Comparison of the critical points (a) and critical exponents (b) of the $AB$ system obtained from series expansions up to order $N=24$ with different transformations, T1 $(엯)$ and T2 $(+)$. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ are given by Eq. (31) in (a) and by $\beta$ at the homogeneous critical point, $\beta =0.2769$, from series expansions [14] for the CP, in (b).

Figure 3

(Color online) Comparison of the critical values (a) and critical exponents $\beta$ (b) of the $AAB$ system obtained from series expansions up to order $N=24$ with different transformations, T1 $(엯)$, T2 $(+)$, and T3 $(◇)$. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ are given by Eq. (31) in (a) and by $\beta$ at the homogeneous critical point, $\beta =0.2769$, from series expansions [14] for the CP, in (b).

Figure 4

(Color online) Comparison of critical values (a) and critical exponents (b) of the $AAB$ lattice obtained from series expansions up to overall order $N=24$ which are cut off at order $M=12$ $(+)$ or in which all the terms up to order $M=23$ $(엯)$ are retained. Transformation T2 was used to obtain these critical points. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ are given by Eq. (31) in (a) and by $\beta$ at the homogeneous critical point, $\beta =0.2769$, from series expansions [14] for the CP, in (b).

Figure 5

(Color online) Comparison of critical values (a) and critical exponents (b) of the $AB$ lattice obtained from series expansions up to overall order $N=24$ which are cut off at order $M=12$ $(+)$ or in which all the terms up to order $M=23$ $(엯)$ are retained. Transformations T2 and T1, respectively, were used to obtain these critical points. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ are given by Eq. (31) in (a) and by $\beta$ at the homogeneous critical point, $\beta =0.2769$, from series expansions [14] for the CP, in (b).

Figure 6

(Color online) Comparison of phase diagram for the $AB$ system obtained from series expansion up to order $N=12$ $(엯)$ and for the $AABB$ chain obtained from series expansion up to order $N=24$ $(+)$. The dashed line $\left(\text{−}\text{−}\text{−}\right)$ is given by Eq. (31).

Figure 7

(Color online) Critical lines in $\log \text{−}\log$ scale for the $AB$ lattice $(엯)$ and the $AAB$ lattice ($+$ and $◇$) as obtained by PDA’s: The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ and $(\bullet \bullet -\bullet \bullet )$ are given by Eq. (31) for the $AB$ and the $AAB$ system, respectively.

Figure 8

(Color online) Comparison of critical values (a) and critical exponents (b) of the disordered system with $p=0.5$ obtained from series expansions up to order $N=12$ using transformations T1 $(엯)$ and T2 $(+)$. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ denote (a) the curve given by Eq. (31) and (b) the value for the critical exponent at the homogeneous critical point for the CP from series expansion, $\beta =0.2769$ [14].

Figure 9

(Color online) Comparison of critical values (a) and critical exponents (b) of the disordered system with $p=0.7$ obtained from series expansions up to order $N=12$: T1 $(엯)$, T2 $(+)$, and T3 $(◇)$. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ denote (a) the curve given by Eq. (31) and (b) the value for the critical exponent at the homogeneous critical point for the CP from series expansion, $\beta =0.2769$ [14].

Figure 10

(Color online) Comparison of critical values (a) and critical exponents (b) of the disordered $1d$ lattice with $p=0.96$ obtained from series expansions up to order $N=12$ using exact configurational averaging $(엯)$ and approximate configurational averaging $(+)$ with $k_{\max }(12,p)=2$. In both expansions T1 was used. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ are given by Eq. (31) in (a) and by $\beta$ at the homogeneous critical point, $\beta =0.2769$, from series expansions [14] for the CP, in (b).

Figure 11

(Color online) Comparison of critical values (a) and critical exponents (b) of the disordered $1d$ lattice with $p=0.96$ obtained from series expansions of the approximately configurationally averaged survival probability up to order $N=19$ using the transformations T1 $(엯)$ and T2 $(+)$. The dashed lines $\left(\text{−}\text{−}\text{−}\right)$ denote (a) the curve given by Eq. (31) and (b) the value for the critical exponent at the homogeneous critical point for the CP from series expansion, $\beta =0.2769$ [14].

Figure 12

(Color online) Critical lines in $\log$-linear scale for the disordered lattice with $p=0.5$ $(엯)$, $p=0.7$ $(+)$, and $p=0.96$ $(◇)$ as obtained by PDA’s: the three lines (—), $\left(\text{−}\text{−}\text{−}\right)$, and $(\bullet \bullet -\bullet \bullet )$ are given by Eq. (31) for the three disordered systems characterized by the values $p=0.5$, $p=0.7$, and $p=0.96$, respectively.