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 τ=2.3135(5) and Ω=0.40(3), consistent with previous numerical results and the recent four-loop series result of Gracey [Phys. Rev. D 92, 025012 (2015)10.1103/PhysRevD.92.025012].