Explosive site percolation with a product rule

We study the site percolation under Achlioptas process with a product rule in a two-dimensional square lattice. From the measurement of the cluster size distribution P-s, we find that P-s has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the P-s distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in two dimensions. Moreover, we also show that the transition nature from the product rule is not the same as that from a sum rule in two dimensions.