Corrected finite-size scaling in percolation

This Rapid Communication proposes a comprehensive scaling theory for percolation, which clarifies the intrinsic nature of finite-size scaling and effectively addresses the finite-size effects. This theory applies to extensive systems, including especially the explosive percolation. It is suggested that explosive percolation shares the same scaling law as normal percolation, but may suffer from more severe finite-size effects. Remarkably, in contrast to previous studies, relying on the framework of our theory, the present Rapid Communication suggests that for all systems, the universal scaling functions do not depend on the boundary conditions.

Figure 1

Derivative of $P_L$ for the NP-BP (a) and EP-BP (b) models, and the scaled derivative peaks for the NP-BP (c) and EP-BP (d) models. In (a) and (b), the arrows indicate how the derivative peak positions change with increasing $L$. In (a), $p_c$ $=$ 0.5 and 1/$v$ $=$ 3/4; in (b), $p_c$ $=$ 0.526560 and 1/$v$ $=$ 0.97.

Figure 2

Scaled derivative peaks for several systems with FBC and PBC. NP-BP and EP-BP are for $P_L$ and the rest are for $R_L$. The system (SP, 3D) is site percolation on simple cubic lattice. The system (BP, 16$L$ × $L$) is bond percolation on 2D square lattice of rectangular boundary [length (the spanning direction) is 16$L$ and width is $L$]. All other systems are bond percolation on 2D square lattice of square boundary. For clarity, the SP system in Table  (site percolation on 2D square lattice of square boundary) is not shown, but it has almost the same scaled derivative peaks as the BP system. For $P_L$, PBC is at both horizontal and vertical directions, while for $R_L$, PBC is only at the direction(s) not for spanning. For all the plotted systems, $L$ $=$ 256.