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### Precise bond percolation thresholds on several four-dimensional lattices

Zhipeng Xun,
Published: 22 January 2020

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 ${p}_{c}=0.160\phantom{\rule{0.16em}{0ex}}131\phantom{\rule{0.16em}{0ex}}2\left(2\right)$, which confirms previous results (based on other methods), and find a new value ${p}_{c}=0.035\phantom{\rule{0.16em}{0ex}}827\left(1\right)$ for the SC-NN+2NN lattice, which was not studied previously for bond percolation. For the 4D bcc and fcc lattices, we obtain ${p}_{c}=0.074\phantom{\rule{0.16em}{0ex}}212\left(1\right)$ and $0.049\phantom{\rule{0.16em}{0ex}}517\left(1\right)$, which are substantially more precise than previous values. We also find critical exponents $\tau =2.3135\left(5\right)$ and $\mathrm{\Omega }=0.40\left(3\right)$, consistent with previous numerical results and the recent four-loop series result of Gracey [ Phys. Rev. D 92, 025012 (2015)].
Keywords: lattices / bond percolation / cubic / neighbors / nearest / Precise / previous

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