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Finite-size scaling in asymmetric systems of percolating sticks

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

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

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

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.

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In order to obtain $θ_{0}$ with high precision successive slopes between points $L$ and $L / 2$ [17] are calculated. The slope is defined as $(\ln | {〈 n 〉}_{L, r} - n_{c} | - \ln | {〈 n 〉}_{L / 2, r} - n_{c} |) / \ln 2$ . This is approximately $[a_{1} (r) / a_{0} (r) / \ln 2] (1 - 2^{θ_{1} - θ_{0}}) L^{- θ_{1} + θ_{0}} - 1 / ν - θ_{0}$ when $| a_{0} (r) | ≫ | a_{1} (r) | L^{- θ_{1} + θ_{0}}$ . Therefore, successive slopes should fall on a straight line when plotted as a function of $L^{- θ_{1} + θ_{0}}$ with intercept at $- 1 / ν - θ_{0}$ . The intercept is not highly sensitive to the value of $- θ_{1} + θ_{0}$ . Note that exponent $θ_{0}$ can not be calculated using successive slopes if $a_{0} (r)$ is close to zero.
The zeroth-order exponent for $Δ_{L, r}^{2}$ can be calculated using successive slopes for all $r$ because condition $| b_{0} (r) | ≫ | b_{1} (r) | L^{- θ_{1} + θ_{0}}$ is always satisfied for large $L$ . The results for the zeroth-order terms are verified by interpolation through antisymmetrizing data points $(Δ_{L, r}^{2} - Δ_{L, 1 / r}^{2}) / 2$ .

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Finite-size scaling in asymmetric systems of percolating sticks

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