Finite-size scaling in quasi-3D stick percolation

Abstract

This work extends the universal finite-size scaling framework for continuum percolation from two-dimensional (2D) to quasi-three-dimensional (Q3D) stick systems, in which sequentially deposited wires of finite diameter stack vertically on a flat substrate. Using Monte Carlo simulation, the percolation threshold is determined for isotropic Q3D stick systems as $N_c l^2 = 6.850923 \pm 0.00014$, approximately $21.5\%$ above the established 2D value of $5.6373$. The threshold is shown to be independent of the wire diameter-to-length ratio $d/l$, reflecting the scale invariance of the contact topology under sequential deposition. Simulation results indicate that, as with 2D networks, by introducing a nonuniversal metric factor, the spanning probability of Q3D stick percolation on square systems with free boundary conditions falls on the same universal scaling function as that for 2D continuum and lattice percolation. This provides substantiating evidence that Q3D stick percolation falls on the same universal scaling function as that for 2D stick percolation and lattice percolation.

Figures (2)

• Figure 1: Finite-size scaling analysis of the spanning probability for 2D widthless stick percolation. (a) Spanning probability $R(N,L)$ as a function of stick density $N$ for system sizes $L = 32$--$256$, with the dashed line indicating $R = 0.5$. (b) Finite-size estimates $N_{0.5}(L)$ plotted against $L^{-7/4}$; the solid line shows the linear extrapolation to $L \to \infty$ used to obtain an initial estimate of $N_c$. Points for small system sizes (red) deviate from linearity, while the large-$L$ points (blue) used in the fit are well described by Eq. \ref{['eq:Nc_convergence']}. (c) $L(R - 0.5)$ versus $\log_2 L$ for three trial densities near criticality; the flat curve at $N = 5.63735$ identifies the critical density where scale invariance holds. (d) Data collapse onto the universal scaling function $F(x) = R - b_0/L$ with $x = (N - N_c)\,L^{1/\nu}$; the dashed line shows the polynomial fit to Eq. \ref{['eq:polynomial']}. The collapse confirms consistency with 2D random percolation universality and serves as a validation of the simulation and fitting methodology against the results of Li and Zhang LiZhang2009.
• Figure 2: Finite-size scaling analysis of the spanning probability for quasi-3D nanowire networks. (a) Crossing probability $R(N,L)$ as a function of wire density $N$ for system sizes $L = 32$--$256$, with the dashed line indicating $R = 0.5$. (b) Finite-size estimates $N_{0.5}(L)$ plotted against $L^{-7/4}$; the solid line shows the linear extrapolation to $L \to \infty$ used to obtain an initial estimate of $N_c$. (c) $L(R - 0.5)$ versus $\log_2 L$ for three trial densities near criticality; the flat curve at $N = 6.85092$ identifies the critical density where scale invariance holds. (d) Data collapse onto the universal scaling function $F(x) = R - b_0/L$ with $x = (N - N_c)\,L^{1/\nu}$, confirming consistency with 2D random percolation universality. The dashed line shows the polynomial fit to the scaling function.