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)].
Highlights
Percolation, which was introduced by Broadbent and Hammersley [1] in 1957, is one of the fundamental models in statistical physics [2,3]
We find a new value for the bond threshold of the complex-neighborhood lattice, simple (hyper) cubic (SC)-nearest neighbors (NN)+2NN
We leave the sites in the unvisited state when we do not occupy them through an occupied bond. (For site percolation, unoccupied visited sites are blocked from ever being occupied in the future.) The single-cluster growth method is similar to the Leath algorithm [9]
Summary
Percolation, which was introduced by Broadbent and Hammersley [1] in 1957, is one of the fundamental models in statistical physics [2,3]. When increasing p from below, a cluster large enough to span the entire system from one side to the other will first appear at a value pc. This point is called the percolation threshold. Percolation thresholds of many lattices can be found analytically [4,5,6,7], while others must be found numerically. Newman and Ziff [10,11] developed a Monte Carlo algorithm which allows one to calculate quantities such as the cluster-size distribution or spanning probability over the entire
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