Finite-size scaling in asymmetric systems of percolating sticks

Milan Žeželj, Igor Stanković, and Aleksandar Belić

Phys. Rev. E 85, 021101 – Published 1 February 2012

Abstract

We investigate finite-size scaling in percolating widthless stick systems with variable aspect ratios in an extensive Monte Carlo simulation study. A generalized scaling function is introduced to describe the scaling behavior of the percolation distribution moments and probability at the percolation threshold. We show that the prefactors in the generalized scaling function depend on the system aspect ratio and exhibit features that are generic to the whole class of the percolating systems. In particular, we demonstrate the existence of a characteristic aspect ratio for which percolation probability at the threshold is scale invariant and definite parity of the prefactors in the generalized scaling function for the first two percolation probability moments.

Received 21 September 2011

DOI:https://doi.org/10.1103/PhysRevE.85.021101

Authors & Affiliations

Milan Žeželj^*, Igor Stanković, and Aleksandar Belić

Scientific Computing Laboratory, Institute of Physics Belgrade, University of Belgrade, RS-11080 Belgrade, Serbia

^*milan.zezelj@ipb.ac.rs

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Vol. 85, Iss. 2 — February 2012

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Images

Figure 1
Percolation probability function $R_{n, L, r}$ [(a), (b), (c)] and probability distribution function $P_{n, L, r}$ [(d), (e), (f)] for stick percolation on rectangular systems with free boundary conditions and increasing system size from $L = 20$ to 200 for three aspect ratios $r = 0.5, 1$ , and 2. The direction of the increase of $L$ is indicated on graphs. The vertical dashed lines denote the value for the percolation threshold $n_{c}$ , while the horizontal dashed lines [(a), (b), (c)] denote the percolation probability on the infinite systems at the threshold $R_{n_{c}, L \to \infty, r}$ .Reuse & Permissions
Figure 2
The dependence of the average stick percolation density ${〈 n 〉}_{L, r}$ on the system size $L$ and aspect ratio $r$ . The values are obtained from Monte Carlo simulations and calculated using Eq. (2). The values are given for aspect ratios $r = 0.7, 0.8, 0.9, 0.95, 0.98, 1$ (solid lines) and their inverse values $r = 1 / 0.7, 1 / 0.8, 1 / 0.9, 1 / 0.95, 1 / 0.98$ (dashed lines). The horizontal bold line denotes the expected value for the percolation threshold $n_{c}$ . Inset: The same data is shown in logarithmic scale to demonstrate the same power-law convergence of the $r$ and $1 / r$ pairs.Reuse & Permissions
Figure 3
Prefactors (a) and exponents (b) are shown for the two leading-order terms of generalized scaling function for average stick percolation density ${〈 n 〉}_{L, r}$ , Eq. (11). The zeroth-order prefactor is an odd function on a logarithmic scale, i.e., $a_{0} (r) = - a_{0} (1 / r)$ , and the zeroth-order exponent is $- 1 / ν$ (solid lines). The first-order prefactor is an even function, i.e., $a_{1} (r) = a_{1} (1 / r)$ , and the first-order correction to the scaling exponent is $θ_{1} = 0.83 (2)$ for $r = 1$ (dashed lines).Reuse & Permissions
Figure 4
Prefactors (a) and exponents (b) are shown for the two leading-order terms of generalized scaling function for variance $Δ_{L, r}^{2}$ , Eq. (12). The zeroth-order prefactor is even function on a logarithmic scale, i.e., $b_{0} (r) = b_{0} (1 / r)$ , and the zeroth-order exponent is $- 1 / ν$ (solid lines). The first-order prefactor is an odd function, i.e., $b_{1} (r) = - b_{1} (1 / r)$ , and the first-order correction to the scaling exponent is $θ_{1} = 0.80 (5)$ for $r = 1$ (dashed lines). Prefactor $b_{1} (r)$ passes through zero for $r$ between 1/0.9 and 1/0.8 causing higher error bars of $θ_{1}$ .Reuse & Permissions
Figure 5
(a) Prefactors for finite-size scaling of the percolation probability at the percolation threshold are shown for the two leading-order terms. (b) Percolation probability at the threshold for infinitely large system $R_{n_{c}, L \to \infty, r}$ . Points represent Monte Carlo data for stick percolation, while the line represents results of Cardy's model for the lattice percolation. The error bars are much smaller than the size of the symbols for all $r$ .Reuse & Permissions

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