Percolation and Criticality in Hyperuniform Networks

TL;DR

This study investigates the percolation behavior of hyperuniform networks, revealing how structural order influences connectivity thresholds and universality classes, with implications for designing resilient disordered systems.

Contribution

It introduces a detailed analysis of percolation thresholds and critical exponents in hyperuniform networks, highlighting the impact of stealthiness on network connectivity and universality class.

Findings

SHU networks have lower percolation thresholds than Poisson networks.

Percolation threshold decreases with increasing stealthiness parameter $\chi$.

Large-$\chi$ SHU networks share universality class with lattices.

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 $\chi$. We construct Delaunay triangulation networks derived from SHU configurations with varying $\chi$ 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 $\chi$, indicating that global connectivity emerges more readily as short-range order increases. Moreover, we show that SHU networks with large $\chi$ belong to the same universality class as lattices, while Poisson and low-$\chi$ 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.