Percolation and Criticality in Hyperuniform Networks

Abstract

Hyperuniform many-particle systems, which encompass crystals, quasicrystals and certain exotic disordered systems, exhibit an anomalous suppression of density fluctuations on macroscopic length scales relative to those of conventional disordered systems. Here we investigate the percolation behaviors of disordered stealthy hyperuniform systems (SHU), a subclass of hyperuniform configurations for which the structure factor vanishes for a finite range of wavevectors near the origin, with the degree of stealthiness controlled via a parameter $χ$. We construct Delaunay triangulation networks derived from SHU configurations with varying $χ$ as well as Poisson point configurations for the purpose of comparison. We investigate a non-uniform bond percolation process, in which bond occupation probabilities decrease with the Euclidean distance between the connected vertices. In this setting, percolation is induced by varying a tuning parameter $z$. We estimate the percolation thresholds $z_c$ and critical exponents of the networks via finite-size scaling and the Newman-Ziff algorithm. We find that SHU networks exhibit lower percolation thresholds than Poisson networks. Notably, the percolation threshold of SHU networks decreases with the stealthiness parameter $χ$, indicating that global connectivity emerges more readily as short-range order increases. Moreover, we show that SHU networks with large $χ$ belong to the same universality class as lattices, while Poisson and low-$χ$ systems show deviations. We relate the shift in critical exponents to the degree of suppression of density fluctuations in the point configurations. Our work extends previous studies on transport properties of SHU systems from continuum two-phase media to networks. These results open new avenues for optimizing the resilience of statistically homogeneous disordered networks.

Figures (7)

• Figure 1: Delaunay triangulation of Poisson and stealthy hyperuniform point configurations. The vertices are shown as dark blue points, and edges are shown in red. Only a section of the periodic network (toroidal topology) is displayed for visual representation.
• Figure 2: The derivative of the wrapping probability, $\frac{d\langle R_{L} (z)\rangle}{dz}$, of stealthy hyperuniform systems with $\chi=0.49$ for $L=20, 50$ and $100$. The error bars on the mean have two sources: (a) intrasample variation, arising from the binomial uncertainty within each realization (negligible in this case due to the large number of runs), and (b) intersample variation, arising from fluctuations across independent realizations, which dominate the reported error bars. The pseudocritical points $z_c(L)$ estimated from Gaussian fitting are marked by blue $\textcolor{blue}{\times}$, whereas quadratic fits near absolute maxima yield estimates marked as red $\textcolor{red}{\times}$. The difference between the two estimates decreases with increasing linear size.
• Figure 3: Extrapolation of the pseudocritical point $z_c(L)$ to the thermodynamic limit ($\Delta z\rightarrow 0$) for all systems. The more ordered systems are found to have a lower critical point. The uncertainties in $\Delta z$ reflect the cross-realization fluctuations of the crossing points $z_1$ and $z_2$. The uncertainties in $z_c(L)$ and $\Delta z$ decrease with the linear size $L$.
• Figure 4: Finite-size scaling of the transition width for all systems. Data points are vertically shifted so that the values at $L=20$ coincide. The expected scaling for the lattice universality class ($\nu=\tfrac{4}{3}$) is shown in cyan for comparison. The linear fits exhibit excellent convergence, supporting the robustness of the critical exponent $\nu$ estimates. Stealthy hyperuniform (SHU) systems with $\chi=0.49$ and $\chi=0.40$ show strong agreement with the lattice (clean) universality class value, while SHU with $\chi=0.20$ and Poisson exhibit deviations.
• Figure 5: The data collapse of stealthy hyperuniform systems with $\chi=0.49$ is shown (a) before and (b) after applying the optimization algorithm described in Sec. \ref{['subsec:opt_refinement']}. In (a), the collapse exhibits a small but visible misalignment, reflecting a deviation from the true $z_c$. After optimization (b), the collapse improves significantly, producing a more accurate estimate of $z_c$.