Finite size scaling theory for percolation phase transition

The finite-size scaling theory for continuous phase transition plays an important role in determining critical point and critical exponents from the size-dependent behaviors of quantities in the thermodynamic limit. For percolation phase transition, the finite-size scaling form for the reduced size of largest cluster has been extended to cluster ranked $R$. However, this is invalid for explosive percolation as our results show. Besides, the behaviors of largest increase of largest cluster induced by adding single link or node have also been used to investigate the critical properties of percolation and several new exponents $\beta_1$, $\beta_2$, $1/\nu_1$ and $1/\nu_2$ are defined while their relation with $\beta/\nu$ and $1/\nu$ is unknown. Through the analysis of asymptotic properties of size jump behaviors, we obtain correct critical exponents and develop a new approach to finite size scaling theory where sizes of ranked clusters are averaged at same distances from the sample-dependent pseudo-critical point in each realization rather than averaging at same value of control parameter.