Explosive Percolation is Continuous, but with Unusual Finite Size Behavior
Physical Review Letters, 106 (22):
225701+ (May 2011)DOI: 10.1103/physrevlett.106.225701
Abstract
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 smax/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-η with η>0. We find different positive values of β (defined via ⟨smax/N⟩∼(p-pc)β for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent Θ (defined such that observables are homogeneous functions of (p-pc)NΘ) is close to—or even equal to—1/2 for all models.