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 $N_c l^2 = 6.850923 \pm 0.000014$, 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 for 2D networks, the spanning probability of Q3D stick percolation on square systems with free boundary conditions collapses onto a single universal scaling function once a nonuniversal metric factor is introduced. This collapse is consistent with the universal scaling function established for 2D continuum and lattice percolation.