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Finite-size scaling in stick percolation

¹School of Information and Communication Technology, Royal Institute of Technology (KTH), Electrum 229, SE-164 40 Kista, Sweden
²The Ångström Laboratory, Uppsala University, P.O. Box 534, SE-751 21 Uppsala, Sweden

^*Corresponding author: shili.zhang@angstrom.uu.se

Phys. Rev. E 80, 040104(R) – Published 19 October, 2009

DOI: https://doi.org/10.1103/PhysRevE.80.040104

Abstract

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 $N_{c} l^{2} = 5.637 26 \pm 0.000 02$ , with $N_{c}$ as the critical density and $l$ as the stick length. Simulation results indicate that by introducing a nonuniversal metric factor $A = 0.106 910 \pm 0.000 009$ , 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.

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References (27)

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The registration can be very simple. For example, in this work we index a subcell as $(i, j)$ with $i + 1$ being its row number and $j + 1$ being its column number and register a stick centered on $(X, Y)$ into the subcell $(i, j)$ if floor $(X) = i$ and floor $(Y) = j$ where the function floor $(x)$ returns the largest integer no greater than $x$ . According to this rule, a stick even centered exactly on a subcell boundary or corner can also be registered into a specific subcell. In addition, we add a row of auxiliary subcells to the bottom and a column of auxiliary subcells to the right-hand side of the system. Then actually there are totally $(L + 1) \times (L + 1)$ subcells so that sticks centered on the bottom or right boundary or the bottom-right corner of the system can also be registered into available subcells.
The time is estimated as follows. For a system of size $L$ and stick density $N$ , there are in total $n = N L^{2}$ sticks and on average each subcell contains $N$ sticks. Each stick may meet $9 N - 1$ sticks at the worst case; these sticks come from the same subcell and the eight neighboring subcells. So, the whole process should check the connectivity between sticks for $(9 N - 1) N L^{2} / 2$ times, i.e., it takes time $O (N^{2} L^{2})$ . Since in large-size stick systems, $N$ of interest is usually less than 10. Hence, the time can be approximated as $O (N L^{2}) = O (n)$ . Note that it might be possible to improve the efficiency a little, at the expense of increased complexity, by using smaller subcells and considering more surrounding subcells as the neighbors. A similar algorithm is discussed by R. L. C. Vink and T. Schilling, Phys. Rev. E 71, 051716 (2005).
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Finite-size scaling in stick percolation

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