Woosik Choi

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 PP(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.

Percolation in spatial evolutionary prisoner's dilemma game on two-dimensional lattices.

We study the spatial evolutionary prisoners dilemma game with updates of imitation max on triangular, hexagonal, and square lattices. We use the weak prisoners dilemma game with a single parameter b. Due to the competition between the temptation value b and the coordination number z of the base lattice, a greater variety of percolation properties is expected to occur on the lattice with the larger z. From the numerical analysis, we find six different regimes on the triangular lattice (z=6). Regardless of the initial densities of cooperators and defectors, cooperators always percolate in the steady state in two regimes for small b. In these two regimes, defectors do not percolate. In two regimes for the intermediate value of b, both cooperators and defectors undergo percolation transitions. The defector always percolates in two regimes for large b. On the hexagonal lattice (z=3), there exist two distinctive regimes. For small b, both the cooperators and the defectors undergo percolation transitions while only defectors always percolate for large b. On the square lattice (z=4), there exist three regimes. Combining with the finite-size scaling analyses, we show that all the observed percolation transitions belong to the universality class of the random percolation. We also show how the detailed growth mechanism of cooperator and defector clusters decides each regime.