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We show analytically that the $[0,1]$, $[1,1]$ and $[2,1]$ Pad{\'e}approximants of the mean cluster number $S(p)$ for site and bond percolation ongeneral $d$-dimensional lattices are upper bounds on this quantity in anyEuclidean dimension $d$, where $p$ is the occupation probability. These resultslead to certain lower bounds on the percolation threshold $p_c$ that becomeprogressively tighter as $d$ increases and asymptotically exact as $d$ becomeslarge. These lower-bound estimates depend on the structure of the$d$-dimensional lattice and whether site or bond percolation is beingconsidered. We obtain explicit bounds on $p_c$ for both site and bondpercolation on five different lattices: $d$-dimensional generalizations of thesimple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices aswell as the $d$-dimensional generalizations of the diamond and kagom{\'e} (orpyrochlore) non-Bravais lattices. These analytical estimates are used to assessavailable simulation results across dimensions (up through $d=13$ in somecases). It is noteworthy that the tightest lower bound provides reasonableestimates of $p_c$ in relatively low dimensions and becomes increasinglyaccurate as $d$ grows. We also derive high-dimensional asymptotic expansionsfor $p_c$ for the ten percolation problems and compare them to theBethe-lattice approximation. Finally, we remark on the radius of convergence ofthe series expansion of $S$ in powers of $p$ as the dimension grows.

Effect of Dimensionality on the Percolation Thresholds of Various $d$-Dimensional Lattices