Finite-size scaling in stick percolation

This work presents the generalization of the concept of universal finite-size scaling functions to continuum percolation. A high-efficiency algorithm for Monte Carlo simulations is developed to investigate, with extensive realizations, the finite-size scaling behavior of stick percolation in large-size systems. The percolation threshold of high precision is determined for isotropic widthless stick systems as $N_c{l}^2=5.63726\pm 0.00002$, with $N_c$ as the critical density and $l$ as the stick length. Simulation results indicate that by introducing a nonuniversal metric factor $A=0.106910\pm 0.000009$, the spanning probability of stick percolation on square systems with free boundary conditions falls on the same universal scaling function as that for lattice percolation.

Figure 1

(Color online) Schematic illustration of stick percolation on a square system $\left(L=5\right)$. Each stick is of unity length $l=1$ and described by its center site and orientation. For clarity, most sticks are presented here only by their centers (black dots). The two interesting system boundaries (the left and right ones) are described also by $L$ connecting sticks. The system is divided into $L\times L$ subcells (dashed lattices) with unity length $l$. Each stick is registered in the subcell where its center lies. It is explicitly shown that a stick in a subcell (with solid boundaries) is impossible to intersect any sticks at other subcells than the same one or its neighbors (the gray ones or light cyan ones).

Figure 2

(Color online) Monte Carlo simulation results for stick percolation on square systems with free boundary conditions. (a) The spanning probability $R$ as a function of stick density $N$ for different system sizes $L$. The horizontal dashed line represents $R=0.5$. (b) The estimated critical density $N_{0.5}\left(L\right)$, for $L=32$, 36, 40, 48, 64, 128, and 256, as a function of $L^{-1-1/v}$. (c) Plot of $L\left(R-0.5\right)$ against ${\text{log}}_2\left(L\right)$ for $N$ near $N_c$, with $L=4$, 8, 16, 32, 64, 128, and 256. The dashed curves represent $R\left(N,L\right)$ given by Eq. (2), neglecting higher orders than $x$, with $b_0=-0.107$ and $a_1=0.107$. (d) Finite-size scaling plot of $R$ after finite-size corrections. $N_c=5.63726$.