Fig. 4: Finite-size scaling in the ring CP model and C-particle networks.

a–d MC simulation results of the ring CP model with different particle numbers N. a Mean degree 〈k〉 approaches the reduced density \(\eta \equiv n{v}_{{{{\rm{ex}}}}}^{{{{\rm{ring}}}}}\) as N increases^39,40. Lines represent empirical fits of the form \(\eta={p}_{1}\langle k\rangle+{p}_{2}{\langle k\rangle }^{{p}_{3}}\) (p₁, p₂, and p₃ are in Supplementary Table 2), used to calculate η_eff(〈k〉; N) in (e, f). b The intersection of S₂/S₁ gives the percolation threshold η_c = 2.11. Inset: the zoom-in at the intersection. Data collapses of S₁ (empty markers), S₂ (filled markers) (c) and χ (d) using η_c from (b) and \((\beta,\gamma,\bar{\nu })=(0.41,1.80,2.64)\) of the standard 3D percolation²³. In DEM simulations of lifted C-particles (θ = 20°), FSS with the same critical exponents and η_c yields reasonable data collapses of S₁ (empty markers), S₂ (filled markers) (e), and χ (f), especially for η_eff ≲ η_c. Markers indicate ensemble averages, and shaded areas indicate standard error bands. Data without FSS are shown in Supplementary Fig. 7.