Precise bond percolation thresholds on several four-dimensional lattices

TL;DR

This study precisely determines bond percolation thresholds on various four-dimensional lattices, including new results for some lattices, and confirms critical exponents consistent with theoretical predictions.

Contribution

The paper provides highly accurate bond percolation thresholds for several 4D lattices, including new values for the SC-NN+2NN lattice, and confirms critical exponents aligning with recent theoretical work.

Findings

Precise bond percolation thresholds for 4D lattices

New threshold value for SC-NN+2NN lattice

Critical exponents consistent with four-loop series results

Abstract

We study bond percolation on several four-dimensional (4D) lattices, including the simple (hyper) cubic (SC), the SC with combinations of nearest neighbors and second nearest neighbors (SC-NN+2NN), the body-centered cubic (BCC), and the face-centered cubic (FCC) lattices, using an efficient single-cluster growth algorithm. For the SC lattice, we find $p_c = 0.1601312(2)$, which confirms previous results (based on other methods), and find a new value $p_c=0.035827(1)$ for the SC-NN+2NN lattice, which was not studied previously for bond percolation. For the 4D BCC and FCC lattices, we obtain $p_c=0.074212(1)$ and 0.049517(1), which are substantially more precise than previous values. We also find critical exponents $\tau = 2.3135(5)$ and $\Omega = 0.40(3)$, consistent with previous numerical results and the recent four-loop series result of Gracey [Phys. Rev. D 92, 025012, (2015)].