Finite size scaling theory for percolation with multiple giant clusters

Abstract

A approach of finite size scaling theory for discontinous percolation with multiple giant clusters is developed in this paper. The percolation in generalized Bohman-Frieze-Wormald (BFW) model has already been proved to be discontinuous phase transition. In the evolution process, the size of largest cluster $s_1$ increases in a stairscase way and its fluctuation shows a series of peaks corresponding to the jumps of $s_1$ from one stair to another. Several largest jumps of the size of largest cluster from single edge are studied by extensive Monte Carlo simulation. $\overline{\Delta}_k(N)$ which is the mean of the $k$th largest jump of largest cluster, $\overline{r}_k(N)$ which is the corresponding averaged edge density, $\sigma_{\Delta,k}(N)$ which is the standard deviation of $\Delta_k$ and $\sigma_{r,k}(N)$ which is the standard deviation of $r_k$ are analyzed. Rich power law behaviours are found for $\overline{r}_k(N)$, $\sigma_{\Delta,k}(N)$ and $\sigma_{r,k}(N)$ with critical exponents denoted as $1/\nu_1$, $(\beta/\nu)_2$ and $1/\nu_2$. Unlike continuous percolation where the exact critical thresholds and critical exponent $1/\nu_1$ are used for finite size scaling, the size-dependent pseudo critical thresholds $\overline{r}_k(N)$ and $1/\nu_2$ works for the data collapse of the curves of largest cluster and its fluctuation in discontinuous percolation in BFW model. Further, data collapse can be obtained part by part. That is, $s_1(r,N)$ can be collapsed for each jump from one stair to another and its fluctuation can be collapsed around each peak with the corresponding $\overline{r}_k(N)$ and $1/\nu_2$.