Dimensional dependence of phase transitions in explosive percolation

To understand the dependence of phase-transition natures in explosive percolations on space dimensions, the number ncut of cutting bonds (sites) and the fractal dimension dCSC of the critical spanning cluster (CSC) for the six different models introduced in Phys. Rev. E 86, 051126 (2012)PLEEE81539-375510. 1103/PhysRevE.86.051126 are studied on two- and three-dimensional lattices. It is found that ncut(L→)=1 for the intrabond-enhanced models and the site models on the two-dimensional square lattice with lattice size L. In contrast, ncut for the intrabond-suppressed models scales as ncut≃Ldcut with dcut=1. dCSC=2.00(1) is obtained for the intrabond-enhanced models and the site models, while dCSC=1.96(1)(<2) is obtained for the intrabond-suppressed models in two dimensions (2D). These results strongly support that the intrabond-enhanced models and the site models undergo the discontinuous transition in 2D, while the intrabond-suppressed models do the continuous transition in 2D. On the three-dimensional cubic lattice, we find that dcut>0 and dCSC=2.8(1)(<3) for all six models, which indicates that the models undergo the continuous transition. Based on the finite-size scaling analyses of mean cluster size and order parameter, all six models in 3D show nearly the same critical phenomena within numerical errors.