Universality class of site and bond percolation on multifractal scale-free planar stochastic lattice

In this article, we investigate both site and bond percolation on a weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network. The characteristic property of percolation is that it exhibits threshold phenomena as we find sudden or abrupt jump in spanning probability across $p_c$ accompanied by the divergence of some other observable quantities, which is reminiscent of a continuous phase transition. Indeed, percolation is characterized by the critical behavior of percolation strength $P\left(p\right)\sim {(p_c-p)}^{\beta }$, mean cluster size $S\sim {(p_c-p)}^{-\gamma }$, and the system size $L\sim {(p_c-p)}^{-\nu }$, which are known as the equivalent counterpart of the order parameter, susceptibility, and correlation length, respectively. Moreover, the cluster size distribution function $n_s\left(p_c\right)\sim s^{-\tau }$ and the mass-length relation $M\sim L^{d_f}$ of the spanning cluster also provide useful characterization of the percolation process. We numerically obtain a value for $p_c$ and for all the exponents such as $\beta ,\nu ,\gamma ,\tau$, and $d_f$. We find that, except for $p_c$, all the exponents are exactly the same in both bond and site percolation despite the significant difference in the definition of cluster and other quantities. Our results suggest that the percolation on WPSL belongs to a new universality class, as its exponents do not share the same value as for all the existing planar lattices. Besides, like all other cases, its site and bond type belong to the same universality class.

Figure 1

A snapshot of the weighted stochastic lattice.

Figure 2

Plots of $log\left[aC\right(a,t\left)\right]$ are shown as function of $a$ for three different times. In the inset we plot $log\left(atC\right(a,t)$ versus $at$ and find that all the distinct plots collapse into one universal curve, revealing that $C(a,t)\sim {\left(at\right)}^{-1}{e}^{-at}$.

Figure 3

The $f\left(\alpha \right)$ spectrum for $m=1.0,1.5,3.5$.

Figure 4

Coordination number distribution of WPSL (or degree distribution function of its dual) $P\left(k\right)$ is the fraction of the total blocks which has coordination number $k$. The slope of the straight line is $\theta =5.56$, revealing that $P\left(k\right)\sim k^{-\theta }$. The plot represents the ensemble average of 500 independent realizations of lattice size $t=50000$.

Figure 5

Spanning probability $W(p,L)$ vs $p$ in WPSL for (a) bond and (b) site percolation. The simulation result of the percolation threshold is $p_c=0.3457$ for bond and 0.5265 for site. In (c) we plot $log(p-p_c)$ vs $logL$ for both bond and site. The two lines have slopes $1/\nu =0.6117\pm 0.0074$ and $0.6135\pm 0.0038$ for bond and site, respectively. In (d) we plot dimensionless quantities $W$ vs $A_1[(p-p_c)L^{1/\nu }-Z_1]$ and we find an excellent data collapse of all the distinct plots in (a) and (b) if we use $Z_1=0.194$ and $A_1=1$ for bond and $A_1=0.798$ for site.

Figure 6

Percolation probability $P(p,L)$ vs $p$ for (a) bond and (b) site percolation. In (c) we plot $logP$ vs $logL$ using data for fixed value of $(p-p_c)L^{1/\nu }$ and find almost parallel lines with slopes $\beta /\nu =0.1356\pm 0.0005$ and $0.1357\pm 0.0002$ for bond and site, respectively. In (d) we plot $B_{2}P{L}^{\beta /\nu }$ vs $B_1((p-p_c)L^{1/\nu }-Z_2)$ and find excellent data collapse of all the distinct plots of (a) and (b) if we use $B_1=1$, $B_2=1$, and $Z_2=0.225$ for bond and use $B_1=0.69$, $B_2=1.443$, and $Z_2=0.225$ for site.

Figure 7

The mean cluster size $S(p,L)$ vs $p$ for (a) bond and (b) site percolation as a function of system size. In (c) we plot $logS$ vs $logL$ using the size of $S$ for fixed value of $(p-p_c)L^{1/\nu }$ and find almost parallel lines with slopes $\gamma /\nu =1.7315\pm -0.0019$ and $1.7280\pm 0.0019$ for bond and site, respectively. In (d) we plot $C_{2}S{L}^{-\gamma /\nu }$ vs $C_1[(p-p_c)L^{1/\nu }-Z_3]$ and find that all distinct plots of (a) and (b) collapse into one universal curve if we use $C_1=1$, $C_2=1$, $Z_3=0.40$ for bond and $C_1=0.625$, $C_2=1.10$, $Z_3=0.40$ for site.

Figure 8

We plot (a) the cluster size distribution function $log\left[n_s\left(p_c\right)\right]$ vs $logs$ for different sizes of the WPSL and find almost parallel lines with slopes 2.07252 and 2.0728 for bond and site percolation, respectively. (b) The mass of the spanning cluster $M$ is shown as a function of system size $L$. The two lines with slopes $d_f=1.8637\pm 0.0224$ and $1.8643\pm 0.0014$ for bond and site, respectively, once again reveal that the fractal dimension of the spanning cluster is independent of the type of percolation.