# Effect of dimensionality on the percolation thresholds of various d-dimensional lattices

## Author(s): Torquato, Salvatore; Jiao, Y

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 Abstract: We show analytically that the [0,1], [1,1], and [2,1] Padé approximants of the mean cluster number S(p) for site and bond percolation on general d-dimensional lattices are upper bounds on this quantity in any Euclidean dimension d, where p is the occupation probability. These results lead to certain lower bounds on the percolation threshold pc that become progressively tighter as d increases and asymptotically exact as d becomes large. These lower-bound estimates depend on the structure of the d-dimensional lattice and whether site or bond percolation is being considered. We obtain explicit bounds on pc for both site and bond percolation on five different lattices: d-dimensional generalizations of the simple-cubic, body-centered-cubic, and face-centered-cubic Bravais lattices as well as the d-dimensional generalizations of the diamond and kagomé (or pyrochlore) non-Bravais lattices. These analytical estimates are used to assess available simulation results across dimensions (up through d=13 in some cases). It is noteworthy that the tightest lower bound provides reasonable estimates of pc in relatively low dimensions and becomes increasingly accurate as d grows. We also derive high-dimensional asymptotic expansions for pc for the 10 percolation problems and compare them to the Bethe-lattice approximation. Finally, we remark on the radius of convergence of the series expansion of S in powers of p as the dimension grows. Publication Date: Mar-2013 Electronic Publication Date: 22-Mar-2013 Citation: Torquato, S, Jiao, Y. (2013). Effect of dimensionality on the percolation thresholds of various d-dimensional lattices. Physical Review E, 87 (3), 10.1103/PhysRevE.87.032149 DOI: doi:10.1103/PhysRevE.87.032149 ISSN: 1539-3755 EISSN: 1550-2376 Pages: 032149-1 - 032149-16 Type of Material: Journal Article Journal/Proceeding Title: Physical Review E Version: Final published version. Article is made available in OAR by the publisher's permission or policy.