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

We show analytically that the [0,1], [1,1], and [2,1] Pade 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 p(c) 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 p(c) 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 kagome (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 p(c) in relatively low dimensions and becomes increasingly accurate as d grows. We also derive high-dimensional asymptotic expansions for p(c) 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. DOI: 10.1103/PhysRevE.87.032149