Gaussian model of explosive percolation in three and higher dimensions

The Gaussian model of discontinuous percolation, recently introduced by Araújo and Herrmann [Phys. Rev. Lett. 105, 035701 (2010)], is numerically investigated in three dimensions, disclosing a discontinuous transition. For the simple cubic lattice, in the thermodynamic limit we report a finite jump of the order parameter $J=0.415\pm 0.005$. The largest cluster at the threshold is compact, but its external perimeter is fractal with fractal dimension $d_A=2.5\pm 0.2$. The study is extended to hypercubic lattices up to six dimensions and to the mean-field limit (infinite dimension). We find that, in all considered dimensions, the percolation transition is discontinuous. The value of the jump in the order parameter, the maximum of the second moment, and the percolation threshold are analyzed, revealing interesting features of the transition and corroborating its discontinuous nature in all considered dimensions. We also show that the fractal dimension of the external perimeter, for any dimension, is consistent with the one from bridge percolation and establish a lower bound for the percolation threshold of discontinuous models with a finite number of clusters at the threshold.

Figure 1

Snapshots of (a) classical percolation and (b) the Gaussian model of discontinuous percolation, at the percolation threshold, on a simple cubic lattice with ${128}^3$ sites. To enhance visibility, the front cube of size ${64}^3$ has been left out in both pictures. While in the classical case clusters have very different size and a fractal shape, for the Gaussian model clusters are rather compact and with a characteristic size. The seven largest clusters of the configurations are shown in (c) for classical percolation and in (d) for the Gaussian model.

Figure 2

System size dependence of the maximum jump of the order parameter $J$ ($\blacksquare$) for the Gaussian model of discontinuous percolation on the simple cubic lattice. In the classical case the change in the order parameter shrinks with the system size and is zero in the thermodynamic limit (not shown), whereas in the discontinuous case a finite, nonzero value ($J=0.415\pm 0.005$) is obtained in this limit. Note that whereas for the product rule and classical percolation the size of the jump decreases with the system size, for the Gaussian model it even slightly increases for the same range of system sizes. Results have been averaged over ${10}^6$ samples for the smallest system size and $1.1\times {10}^3$ samples for the largest one. Random numbers have been generated with the algorithm proposed in Ref. [20]. To identify the clusters and keep track of their properties we have considered the labeling scheme proposed by Newman and Ziff [21], related to the Hoshen–Kopelman algorithm [22].

Figure 3

Threshold $p_c$ as a function of the inverse linear system size $L^{-a}$ for the Gaussian model of discontinuous percolation on the simple cubic lattice. $p_{c,J}$ $(\times )$ stands for the average fraction of occupied bonds at which the jump occurs and $p_{c,M}$ $(+)$ for the position of the maximum of the second moment of the cluster-size distribution without the contribution of the largest cluster. The percolation threshold is estimated to be $0.3468\pm 0.0005$. Results have been averaged over ${10}^3$ samples for the smallest system size (${32}^3$ sites) and ${10}^2$ samples for the largest one (${256}^3$ sites). The inset shows the same for the case where only merging bonds are considered; that is, all clusters are trees (loopless). In this case the threshold estimators [$p_{c,J}$ $(•)$, $p_{c,M}$ $\left(▴\right)$] depend asymptotically linearly on $L^{-3}$. The threshold for the loopless case is estimated to be $0.3333\pm 0.0004$. Results have been averaged over ${10}^3$ samples for the smallest system site (${50}^3$ sites) and ${10}^2$ samples for the largest one (${512}^3$ sites). Error bars are smaller than the symbol size.

Figure 4

Maximum of the second moment of the cluster-size distribution per lattice site, $M_2^{\prime }\left(p_{c,M}\right)/L^d$ ($*$), as a function of the linear system size $L$. The second moment per lattice site tends toward a constant value, as expected for a discontinuous transition. Results have been averaged over ${10}^3$ samples for the smallest system size and $2.4\times {10}^2$ samples for the largest one.

Figure 5

Standard deviation of the order parameter ${\chi }_{\infty }$ as a function of the bond occupation fraction $p$ for different linear system sizes $L$. One observes that the peak increases and narrows with the system size. In the inset we see ${\chi }_{\infty }$ as a function of the scaling variable $(p-p_c)L^a$, with $a=1.64$, for different linear system sizes $L$: 16 ($+$), 32 ($\times$), 64 ($*$), 128 ($\square$), 256 ($•$), and 512 ($\blacksquare$). Results have been averaged over ${10}^8$ samples for the smallest system size and 28 samples for the largest one.

Figure 6

System size dependence of the jump $J$ for the Gaussian model of discontinuous percolation on the hypercubic lattice of dimension 3 ($\times$), 4 ($⧫$), 5 ($▴$), and 6 ($•$), as well as in the mean-field case ($*$). In the limit $N^{-1}\to 0$, the jump has within the error bars the same finite value $0.415$ for all considered graphs. The solid line is a guide to the eye and of the form $0.415-40N^{-1}$. For the sake of comparison we plot the jump as a function of the inverse system size $N^{-1}$. Results have been averaged over ${10}^7$ samples for the smallest system size and at least 10 samples for the largest one.

Figure 7

Cluster-size distribution for the Gaussian model on the simple cubic lattice. The fraction $n_s$ ($•$) of clusters of size $s$ times the size of the cluster $s/N$ is shown as a function of $s/N$. One observes that the distribution is bimodal, as expected for discontinuous transitions. The system size is ${64}^3$ sites, results have been averaged over $1.6\times {10}^5$ samples, and error bars are indicated.

Figure 8

System size dependence of the external perimeter of the largest cluster $A$, at the threshold, for the Gaussian model of discontinuous percolation on the hypercubic lattice of dimension 3 ($*$), 4 ($⧫$), 5 ($•$), and 6 ($\blacksquare$). One observes that $A$ asymptotically scales with the system size as $A\sim L^{d_A}$. The solid lines have slopes of $5.9\pm 0.8$, $4.9\pm 0.7$, $3.6\pm 0.4$, and $2.5\pm 0.2$, respectively. Results have been averaged over ${10}^3$ samples.