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

S. Torquato*

Y. Jiao†

  • Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

  • Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton New Jersey 08544, USA

  • *torquato@electron.princeton.edu
  • †yjiao@princeton.edu
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Phys. Rev. E 87, 032149 – Published 22 March, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.032149

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.

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Phys. Rev. E 87, 032149– Published 22 March, 2013
  • Received 1 February 2013
Article part of CHORUS

©2013 American Physical Society

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