Precise bond percolation thresholds on several four-dimensional lattices

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)].

References in corpus (6)

• Square lattice site percolation at increasing ranges of neighbor interactions
• Exact bond percolation thresholds in two dimensions
• Correction-to-scaling exponent for two-dimensional percolation
• Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors
• Percolation of overlapping squares or cubes on a lattice
• Spreading with immunization in high dimensions