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Higher-order orientational effects on geometric percolation in two-dimensional stick networks

Davide Grazioli1,*, Angelo Simone1,†, Aidan Durant2,‡, Avik P. Chatterjee2,3,§, and Claudio Grimaldi4,5,∥

  • *Contact author: davide.grazioli@unipd.it
  • †Contact author: angelo.simone@unipd.it
  • ‡Contact author: adurant@esf.edu
  • §Contact author: achatter@esf.edu
  • ∥Contact author: claudio.grimaldi@epfl.ch
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Phys. Rev. E 114, 044303 – Published 1 October, 2026

DOI: https://doi.org/10.1103/43yn-sh6g

Abstract

We investigate geometric percolation in two-dimensional assemblies of randomly positioned, widthless sticks of uniform length. The stick orientations are sampled from probability density functions with a vanishing mean orientational order parameter, while their higher-order structure varies continuously from perfect isotropy to perfect cross-alignment. Monte Carlo simulations show that the percolation threshold is lowest for the isotropic distribution and increases toward the perfectly cross-aligned limit. This variation is strongly nonuniform: small orientational perturbations near perfect cross-alignment produce substantially larger changes in the threshold than comparable perturbations near isotropy. The simulation results are interpreted using an orientation-resolved eigenvalue formulation based on the full excluded-area kernel and a fully preaveraged excluded-area approximation. For the probability density functions considered, the two formulations predict nearly identical thresholds. Orientation-dependent variations in the expected contact numbers therefore provide only a minor correction, whereas the dominant orientational dependence is retained by the averaged excluded area between pairs of sticks. A harmonic representation of this quantity further shows that the second moment captures the leading variation near isotropy, while progressively higher-order moments are required as the distribution approaches perfect cross-alignment. The results identify the orientational information that must be retained to predict geometric percolation in distributions having the same mean order parameter but different higher-order structures.

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Phys. Rev. E 114, 044303– Published 1 October, 2026
  • Received 3 August 2026
  • Accepted 9 September 2026
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