Finite-size scaling theory for explosive percolation transitions

The finite-size scaling (FSS) theory for continuous phase transitions has been useful in determining the critical behavior from the size-dependent behaviors of thermodynamic quantities. When the phase transition is discontinuous, however, FSS approach has not been well established yet. Here, we develop a FSS theory for the explosive percolation transition arising in the Erdos and Rényi model under the Achlioptas process. A scaling function is derived based on the observed fact that the derivative of the curve of the order parameter at the critical point t_c diverges with system size in a power-law manner, which is different from the conventional one based on the divergence of the correlation length at t_c. We show that the susceptibility is also described in the same scaling form. Numerical simulation data for different system sizes are well collapsed on the respective scaling functions.