Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods

TL;DR

This study investigates how the percolation threshold behaves on four-dimensional hypercubic lattices with extended neighborhoods, confirming theoretical predictions and analyzing finite-size corrections through high-precision Monte Carlo simulations.

Contribution

It provides detailed numerical analysis of percolation thresholds with extended neighborhoods in four dimensions, confirming asymptotic behaviors and finite-size correction models for both site and bond percolation.

Findings

For site percolation, confirms $zp_c o 16 \, \eta_c$ as $z$ increases.

Finite-size corrections for site percolation fit models involving $z+b$ and exponential forms.

For bond percolation, finite-$z$ correction aligns with $zp_c - 1 \sim a_1 (\ln z)/z$, with alternative behavior not excluded.

Abstract

The asymptotic behavior of the percolation threshold $p_c$ and its dependence upon coordination number $z$ is investigated for both site and bond percolation on four-dimensional lattices with compact extended neighborhoods. Simple hypercubic lattices with neighborhoods up to 9th nearest neighbors are studied to high precision by means of Monte-Carlo simulations based upon a single-cluster growth algorithm. For site percolation, an asymptotic analysis confirms the predicted behavior $zp_c \sim 16 \eta_c = 2.086$ for large $z$, and finite-size corrections are accounted for by forms $p_c \sim 16 \eta_c/(z+b)$ and $p_c \sim 1- \exp(-16 \eta_c/z)$ where $\eta_c \approx 0.1304$ is the continuum percolation threshold of four-dimensional hyperspheres. For bond percolation, the finite-$z$ correction is found to be consistent with the prediction of Frei and Perkins, $zp_{c} - 1 \sim a_{1} (\ln z)/z$, although the behavior $zp_{c} - 1 \sim a_1 z^{-3/4}$ cannot be ruled out.