Mapping functions and critical behavior of percolation on rectangular domains

The existence probability $E_p$ and the percolation probability $P$ of bond percolation on rectangular domains with different aspect ratios $R$ are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of $E_p$ and $P$ for such systems with exponents $a$ and $b$, respectively, found by Watanabe et al. [Phys. Rev. Lett. 93, 190601 (2004)] can be understood from the lower-order approximation of the mapping functions $f_R$ and $g_R$ for $E_p$ and $P$, respectively; the exponents $a$ and $b$ can be obtained from numerically determined mapping functions $f_R$ and $g_R$, respectively.

Figure 1

Mapping functions of the bond percolation model on square lattices with free boundary conditions. (a) $f_Q$ for the existence probabilities $E_p$ as a function of $\rho$, and (b) $g_Q$ for the percolation probabilities $P$ as a function of $\rho$. The insets in (a) and (b) show how to construct the mapping functions for $E_p$ and $P$, respectively.

Figure 2

UFSSFs of (a) existence probabilities and (b) percolation probabilities corresponding to Fig. 1. The aspect ratio is $R=1$, and the system sizes are $L=64$ $(Q=1)$, 128 $(Q=2)$, and 256 $(Q=4)$. The scaling variable is denoted by $\epsilon \equiv (\rho -{\rho }_c)L^{y_t}$.

Figure 3

(a) Procedures to obtain the mapping functions of existence probabilities with different aspect ratios. (b) Obtained mapping functions of systems with $L=256$ and $R=1$, 2, 4, 8, and 16 (from bottom to top). The solid circles are the effective critical points ${\rho }_c^{\prime }$.

Figure 4

(a) Procedures to obtain the mapping functions of percolation probabilities with different aspect ratios. (b) Obtained mapping functions of systems with $L=128$ and $R=1$, 2, 4, 8, and 16 (from bottom to top).

Figure 5

Superscaling plot of the percolation probabilities $P$ using Eq. (23) for systems with $R=1$, 2, 4, 8, and 16, and $L=128$ with $b=0.05$.

Figure 6

Plots of (a) $u={\phantom{\mid }df∕d\rho \mid }_{{\rho }_c^{\prime }}$ and (b) $v={\phantom{\mid }dg∕d\rho \mid }_{{\rho }_c^{\prime }}$ as a function $R$. The solid lines are $R^a$ and $R^b$ with exponents $a=0.14$ and $b=0.05$, respectively. Decimal logarithms are taken for both axes.