Site percolation and random walks on d-dimensional Kagomé lattices

The site percolation problem is studied on d-dimensional generalizations of the Kagomé lattice. These lattices are isotropic and have the same coordination number q as the hyper-cubic lattices in d dimensions, namely q = 2d . The site percolation thresholds are calculated numerically for d = 3, 4, 5, and 6. The scaling of these thresholds as a function of dimension d, or alternatively q, is different than for hypercubic lattices: pc ∼ 2/q instead of pc ∼ 1/(q − 1). The latter is the Bethe approximation, which is usually assumed to hold for all lattices in high dimensions. A series expansion is calculated, in order to understand the different behaviour of the Kagomé lattice. The return probability of a random walker on these lattices is also shown to scale as 2/q. For bond percolation on d-dimensional diamond lattices these results imply pc ∼ 1/(q − 1).