Explosive Percolation is Continuous, but with Unusual Finite Size Behavior

Abstract

We study four Achlioptas type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entire holomorphic. The distributions of the order parameter, the relative size $s_{\rm max}/N$ 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 $< s_{\rm max}/N > \sim (p-p_c)^\beta$ 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.