Percolation on a multifractal scale-free planar stochastic lattice and its universality class

We investigate site percolation on a weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network. Percolation is typically characterized by a threshold value $p_c$ at which a transition occurs and by a set of critical exponents $\beta , \gamma , \nu$ which describe the critical behavior of the percolation probability $P\left(p\right)$, mean cluster size $S\left(p\right)$, and the correlation length $\xi$. Besides, the exponent $\tau$ characterizes the cluster size distribution function $n_s\left(p_c\right)$ and the fractal dimension $d_f$ characterizes the spanning cluster. We numerically obtain the value of $p_c$ and of all the exponents. These results suggest that the percolation on WPSL belong to a separate universality class than on all other planar lattices.

Figure 1

A snapshot of the weighted stochastic lattice.

Figure 2

(a) Plots of spanning probability $W\left(p\right)$ vs $p$ for different lattice sizes $L=N^{1/2}$, where $N=1+3t$ and $t=1000$, 6500, and $66667$. The arrow points in the direction of increasing $L$. The inset shows a linear regression fit to the data points obtained by plotting $ln(p_c-p)$ vs $ln\left(L\right)$ with a slope $1/\nu =0.6112\left(38\right)$, where each data point represents an average over $300000$ independent realizations. (b) Excellent data collapse confirms the correctness of our estimation of the exponent $\nu$.

Figure 3

(a) Plots of percolation probability $P\left(p\right)$ vs $p$ for different sizes $L\left(t\right)$ where the arrow points in the direction of increasing $L\left(t\right)$, and $t=2500$, 6500, 13 500, 22 534, and 66 667. Here, each data point represents an average over $300000$ independent realizations. The inset shows a plot of $ln\left(P\right)$ at a given value of $(p-p_c)L^{1/\nu }$ vs $ln\left(L\right)$. The linear regression fit gives an excellent straight line with a slope $\beta /\nu =0.1357\left(5\right)$. (b) We plot $P{L}^{\beta /\nu }$ vs $(p-p_c)L^{1/\nu }$ instead of $P$ vs $p$ and find an excellent data-collapse which confirms that the estimates for the exponents $\beta$ and $\nu$ are correct.

Figure 4

(a) Mean cluster area $S\left(p\right)$ vs $p$ for different $L$. The arrow points in the direction of increasing $L\left(t\right)$, where $t=2500$, 6500, 13 500, and 22 534. Here, each data point represents an average over $200000$ independent realizations. The inset shows a plot of $ln\left(S\right)$ at a given value of $(p-p_c)L^{1/\nu }$ vs $ln\left(L\right)$. The linear regression yields an excellent straight line with a slope $\gamma /\nu =1.7281\left(19\right)$. (b) We plot $S{L}^{-\gamma /\nu }$ vs $(p-p_c)L^{1/\nu }$ find that all distinct curves of (a) collapsed onto a single universal curve as expected according to finite-scaling hypothesis. It confirms that the estimates for the exponents $\gamma$ and $\nu$ are correct.

Figure 5

(a) The plot of $log\left(n_s\right)$ vs $log\left(s\right)$ for different lattice sizes $L={(1+3t)}^{1/2}$, where $t=2500$, 6500, 13 500, and 22 534. (b) The plot of $ln\left(M\right)$ vs $ln\left(L\right)$. In either case, each data point represents an average over $100000$ independent realizations.