Explosive Percolation is Continuous, but with Unusual Finite Size Behavior

We study four Achlioptas-type processes with “explosive” percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entirely holomorphic. The distributions of the order parameter, i.e., the relative size $s_{\max }/N$ of the largest cluster, are double humped. But—in contrast to first-order phase transitions—the distance between the two peaks decreases with system size $N$ as $N^{-\eta }$ with $\eta >0$. We find different positive values of $\beta$ (defined via $⟨s_{\max }/N⟩\sim (p-p_c{)}^{\beta }$ for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent $\Theta$ (defined such that observables are homogeneous functions of $(p-p_c)N^{\Theta }$) is close to—or even equal to—$1/2$ for all models.

Figure 1

Distributions of the order parameter $s_{\max }/N$ for four EP models. They are shown at the effective critical point, defined such that both peaks have the same height. Normalization is such that their height is 1. For the largest systems, curves were approximated by cubic splines to make them smooth.

Figure 2

Data collapses for the two peaks in the order parameter distribution for the CDGM model. Colors and line styles are the same as in Fig. 1.

Figure 3

Log-log plot of the average order parameter for the CGDM model versus $p-p_c$, for six different values of $N$. One sees clearly a common part with slope $\beta$ (indicated also by the straight line), from which curves for different $N$ deviate later and later, as $N$ increases. The inset shows the collapse of these data as predicted by Eq. (2). While $\Theta$ is fitted, both $\beta$ and $p_c$ are taken from 20.

Figure 4

Doubly linear plot of the same data shown in Fig. 3, but extended to values $p<p_c$. Here we used $\Theta =1/2$, which gives worse data collapse for $p>p_c$, but vastly more systematic behavior for $p<p_c$. The inset shows a blowup of the region around $p=p_c$, with logarithmic $y$ axis. The decrease of the curves at $z=0$ with $N$ suggests that $z_0=0$, and that a new power law holds for $p=p_c$.